advance design 2010 - validation guide
DESCRIPTION
Advance Design 2010 - Validation GuideTRANSCRIPT
Advance Design
Validation Guide
ADVANCE VALIDATION GUIDE
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Contents
1 INTRODUCTION....................................................................................................................................................................1 1.1 Test description documents coding.....................................................................................................................2 1.2 Test description documents coding example......................................................................................................3 1.3 Results comparison documents coding example ...............................................................................................4 1.4 Margin of error........................................................................................................................................................4 1.5 Synthetic sheet of 2010 version............................................................................................................................5
2 DETAILED TESTS DESCRIPTION .......................................................................................................................................9 2.1 Test No. 01-0001SSLSB_FEM: Cantilever rectangular plate ............................................................................10
2.1.1 Description sheet .....................................................................................................................................10 2.1.2 Overview ..................................................................................................................................................10 2.1.3 Displacement of the model in the linear elastic range ..............................................................................11 2.1.4 Results sheet ...........................................................................................................................................11
2.2 Test No. 01-0002SSLLB_FEM: System of two bars with three hinges ............................................................12 2.2.1 Description sheet .....................................................................................................................................12 2.2.2 Overview ..................................................................................................................................................12 2.2.3 Displacement of the model in C ...............................................................................................................13 2.2.4 Bars stresses ...........................................................................................................................................13 2.2.5 Shape of the stress diagram ....................................................................................................................14 2.2.6 Results sheet ...........................................................................................................................................14
2.3 Test No. 01-0003SSLSB_FEM: Circular plate under uniform load ...................................................................15 2.3.1 Description sheet .....................................................................................................................................15 2.3.2 Overview ..................................................................................................................................................15 2.3.3 Vertical displacement of the model at the center of the plate ...................................................................16 2.3.4 Results sheet ...........................................................................................................................................17
2.4 Test No. 01-0004SDLLB_FEM: Slender beam with variable section (fixed-free) ............................................18 2.4.1 Description sheet .....................................................................................................................................18 2.4.2 Overview ..................................................................................................................................................18 2.4.3 Eigen mode frequencies...........................................................................................................................19 2.4.4 Results sheet ...........................................................................................................................................20
2.5 Test No. 01-0005SSLLB_FEM: Tied (sub-tensioned) beam ..............................................................................21 2.5.1 Description sheet .....................................................................................................................................21 2.5.2 Overview ..................................................................................................................................................21 2.5.3 Compression force in CE bar ...................................................................................................................22 2.5.4 Bending moment at point H......................................................................................................................23 2.5.5 Vertical displacement at point D...............................................................................................................24 2.5.6 Results sheet ...........................................................................................................................................25
2.6 Test No. 01-0006SDLLB_FEM: Thin circular ring fixed in two points..............................................................26 2.6.1 Description sheet .....................................................................................................................................26 2.6.2 Overview ..................................................................................................................................................26 2.6.3 Eigen mode frequencies...........................................................................................................................27 2.6.4 Results sheet ...........................................................................................................................................28
2.7 Test No. 01-0007SDLSB_FEM: Thin lozenge-shaped plate fixed on one side (α = 0 °) .................................29 2.7.1 Description sheet .....................................................................................................................................29 2.7.2 Overview ..................................................................................................................................................29 2.7.3 Eigen mode frequencies relative to the α angle .......................................................................................30 2.7.4 Results sheet ...........................................................................................................................................30
2.8 Test No. 01-0008SDLSB_FEM: Thin lozenge-shaped plate fixed on one side (α = 15 °) ................................31 2.8.1 Description sheet .....................................................................................................................................31
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2.8.2 Overview.................................................................................................................................................. 31 2.8.3 Eigen modes frequencies function by α angle ......................................................................................... 32 2.8.4 Results sheet ........................................................................................................................................... 32
2.9 Test No. 01-0009SDLSB_FEM: Thin lozenge-shaped plate fixed on one side (α = 30 °)................................ 33 2.9.1 Description sheet ..................................................................................................................................... 33 2.9.2 Overview.................................................................................................................................................. 33 2.9.3 Eigen mode frequencies relative to the α angle....................................................................................... 34 2.9.4 Results sheet ........................................................................................................................................... 34
2.10 Test No. 01-0010SDLSB_FEM: Thin lozenge-shaped plate fixed on one side (α = 45 °)................................ 35 2.10.1 Description sheet ..................................................................................................................................... 35 2.10.2 Overview.................................................................................................................................................. 35 2.10.3 Eigen mode frequencies relative to the α angle....................................................................................... 36 2.10.4 Results sheet ........................................................................................................................................... 36
2.11 Test No. 01-0011SDLLB_FEM: Vibration mode of a thin piping elbow in plane (case 1) .............................. 37 2.11.1 Description sheet ..................................................................................................................................... 37 2.11.2 Overview.................................................................................................................................................. 37 2.11.3 Eigen mode frequencies .......................................................................................................................... 38 2.11.4 Results sheet ........................................................................................................................................... 38
2.12 Test No. 01-0012SDLLB_FEM: Vibration mode of a thin piping elbow in plane (case 2) .............................. 39 2.12.1 Description sheet ..................................................................................................................................... 39 2.12.2 Overview.................................................................................................................................................. 39 2.12.3 Eigen mode frequencies .......................................................................................................................... 40 2.12.4 Results sheet ........................................................................................................................................... 40
2.13 Test No. 01-0013SDLLB_FEM: Vibration mode of a thin piping elbow in plane (case 3) .............................. 41 2.13.1 Description sheet ..................................................................................................................................... 41 2.13.2 Overview.................................................................................................................................................. 41 2.13.3 Eigen mode frequencies .......................................................................................................................... 42 2.13.4 Results sheet ........................................................................................................................................... 42
2.14 Test No. 01-0014SDLLB_FEM: Thin circular ring hang from an elastic element............................................ 43 2.14.1 Description sheet ..................................................................................................................................... 43 2.14.2 Overview.................................................................................................................................................. 43 2.14.3 Eigen mode frequencies .......................................................................................................................... 44 2.14.4 Results sheet ........................................................................................................................................... 45
2.15 Test No. 01-0015SSLLB_FEM: Double fixed beam with a spring at mid span................................................ 46 2.15.1 Description sheet ..................................................................................................................................... 46 2.15.2 Overview.................................................................................................................................................. 46 2.15.3 Displacement of the model in the linear elastic range.............................................................................. 47 2.15.4 Results sheet ........................................................................................................................................... 48
2.16 Test No. 01-0016SDLLB_FEM: Double fixed beam ........................................................................................... 49 2.16.1 Description sheet ..................................................................................................................................... 49 2.16.2 Overview.................................................................................................................................................. 49 2.16.3 Displacement of the model in the linear elastic range.............................................................................. 50 2.16.4 Eigen mode frequencies of the model in the linear elastic range............................................................. 50 2.16.5 Results sheet ........................................................................................................................................... 52
2.17 Test No. 01-0017SDLLB_FEM: Short beam on simple supports (on the neutral axis) .................................. 53 2.17.1 Description sheet ..................................................................................................................................... 53 2.17.2 Overview.................................................................................................................................................. 53 2.17.3 Eigen modes frequencies ........................................................................................................................ 54 2.17.4 Results sheet ........................................................................................................................................... 55
2.18 Test No. 01-0018SDLLB_FEM: Short beam on simple supports (eccentric) .................................................. 56 2.18.1 Description sheet ..................................................................................................................................... 56 2.18.2 Overview.................................................................................................................................................. 56 2.18.3 Eigen modes frequencies ........................................................................................................................ 57
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2.18.4 Results sheet ...........................................................................................................................................59 2.19 Test No. 01-0019SDLSB_FEM: Thin square plate fixed on one side ...............................................................60
2.19.1 Description sheet .....................................................................................................................................60 2.19.2 Overview ..................................................................................................................................................60 2.19.3 Eigen modes frequencies.........................................................................................................................61 2.19.4 Results sheet ...........................................................................................................................................62
2.20 Test No. 01-0020SDLSB_FEM: Rectangular thin plate simply supported on its perimeter ...........................63 2.20.1 Description sheet .....................................................................................................................................63 2.20.2 Overview ..................................................................................................................................................63 2.20.3 Eigen modes frequencies.........................................................................................................................64 2.20.4 Results sheet ...........................................................................................................................................65
2.21 Test No. 01-0021SFLLB_FEM: Cantilever beam in Eulerian buckling .............................................................66 2.21.1 Description sheet .....................................................................................................................................66 2.21.2 Overview ..................................................................................................................................................66 2.21.3 Critical load on node 5..............................................................................................................................67 2.21.4 Results sheet ...........................................................................................................................................67
2.22 Test No. 01-0022SDLSB_FEM: Annular thin plate fixed on a hub (repetitive circular structure) ..................68 2.22.1 Description sheet .....................................................................................................................................68 2.22.2 Overview ..................................................................................................................................................68 2.22.3 Eigen modes frequencies.........................................................................................................................69 2.22.4 Results sheet ...........................................................................................................................................69
2.23 Test No. 01-0023SDLLB_FEM: Bending effects of a symmetrical portal frame..............................................70 2.23.1 Description sheet .....................................................................................................................................70 2.23.2 Overview ..................................................................................................................................................70 2.23.3 Eigen modes frequencies.........................................................................................................................71 2.23.4 Results sheet ...........................................................................................................................................72
2.24 Test No. 01-0024SSLLB_FEM: Slender beam on two fixed supports ..............................................................73 2.24.1 Description sheet .....................................................................................................................................73 2.24.2 Overview ..................................................................................................................................................73 2.24.3 Shear force at G.......................................................................................................................................75 2.24.4 Bending moment in G...............................................................................................................................76 2.24.5 Vertical displacement at G .......................................................................................................................77 2.24.6 Horizontal reaction at A ............................................................................................................................78 2.24.7 Results sheet ...........................................................................................................................................78
2.25 Test No. 01-0025SSLLB_FEM: Slender beam on three supports.....................................................................79 2.25.1 Description sheet .....................................................................................................................................79 2.25.2 Overview ..................................................................................................................................................79 2.25.3 Bending moment at B...............................................................................................................................80 2.25.4 Reaction in B............................................................................................................................................80 2.25.5 Vertical displacement at B........................................................................................................................81 2.25.6 Results sheet ...........................................................................................................................................81
2.26 Test No. 01-0026SSLLB_FEM: Bimetallic: Fixed beams connected to a stiff element...................................82 2.26.1 Description sheet .....................................................................................................................................82 2.26.2 Overview ..................................................................................................................................................82 2.26.3 Deflection at B and D ...............................................................................................................................83 2.26.4 Vertical reaction at A and C......................................................................................................................84 2.26.5 Bending moment at A and C ....................................................................................................................84 2.26.6 Results sheet ...........................................................................................................................................84
2.27 Test No. 01-0027SSLLB_FEM: Fixed thin arc in planar bending......................................................................85 2.27.1 Description sheet .....................................................................................................................................85 2.27.2 Overview ..................................................................................................................................................85 2.27.3 Displacements at B ..................................................................................................................................86 2.27.4 Results sheet ...........................................................................................................................................87
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2.28 Test No. 01-0028SSLLB_FEM: Fixed thin arc in out of plane bending............................................................ 88 2.28.1 Description sheet ..................................................................................................................................... 88 2.28.2 Overview.................................................................................................................................................. 88 2.28.3 Displacements at B.................................................................................................................................. 89 2.28.4 Moments at θ = 15°.................................................................................................................................. 89 2.28.5 Results sheet ........................................................................................................................................... 89
2.29 Test No. 01-0029SSLLB_FEM: Double hinged thin arc in planar bending...................................................... 90 2.29.1 Description sheet ..................................................................................................................................... 90 2.29.2 Overview.................................................................................................................................................. 90 2.29.3 Displacements at A, B and C ................................................................................................................... 91 2.29.4 Results sheet ........................................................................................................................................... 92
2.30 Test No. 01-0030SSLLB_FEM: Portal frame with lateral connections............................................................. 93 2.30.1 Description sheet ..................................................................................................................................... 93 2.30.2 Overview.................................................................................................................................................. 93 2.30.3 Displacements at A.................................................................................................................................. 94 2.30.4 Moments in A........................................................................................................................................... 95 2.30.5 Results sheet ........................................................................................................................................... 95
2.31 Test No. 01-0031SSLLB_FEM: Truss with hinged bars under a punctual load.............................................. 96 2.31.1 Description sheet ..................................................................................................................................... 96 2.31.2 Overview.................................................................................................................................................. 96 2.31.3 Displacements at C and D ....................................................................................................................... 97 2.31.4 Results sheet ........................................................................................................................................... 97
2.32 Test No. 01-0032SSLLB_FEM: Beam on elastic soil, free ends....................................................................... 98 2.32.1 Description sheet ..................................................................................................................................... 98 2.32.2 Overview.................................................................................................................................................. 98 2.32.3 Bending moment and displacement at C ................................................................................................. 99 2.32.4 Displacements at A................................................................................................................................ 100 2.32.5 Results sheet ......................................................................................................................................... 100
2.33 Test No. 01-0033SFLLA_FEM: EDF Pylon ....................................................................................................... 101 2.33.1 Description sheet ................................................................................................................................... 101 2.33.2 Overview................................................................................................................................................ 101 2.33.3 Displacement of the model in the linear elastic range............................................................................ 103 2.33.4 Results sheet ......................................................................................................................................... 105
2.34 Test No. 01-0034SSLLB_FEM: Beam on elastic soil, hinged ends................................................................ 106 2.34.1 Description sheet ................................................................................................................................... 106 2.34.2 Overview................................................................................................................................................ 106 2.34.3 Displacement and support reaction at A ................................................................................................ 107 2.34.4 Displacement and bending moment at D ............................................................................................... 108 2.34.5 Results sheet ......................................................................................................................................... 109
2.35 Test No. 01-0035SSLPB_FEM: Plate with in plane bending and shear ......................................................... 110 2.35.1 Description sheet ................................................................................................................................... 110 2.35.2 Overview................................................................................................................................................ 110 2.35.3 Planes stresses in (x,y).......................................................................................................................... 110 2.35.4 Results sheet ......................................................................................................................................... 111
2.36 Test No. 01-0036SSLSB_FEM: Simply supported square plate..................................................................... 112 2.36.1 Description sheet ................................................................................................................................... 112 2.36.2 Overview................................................................................................................................................ 112 2.36.3 Vertical displacement at O ..................................................................................................................... 113 2.36.4 Results sheet ......................................................................................................................................... 113
2.37 Test No. 01-0037SSLSB_FEM: Caisson beam in torsion ............................................................................... 114 2.37.1 Description sheet ................................................................................................................................... 114 2.37.2 Overview................................................................................................................................................ 114 2.37.3 Displacement and stress at two points................................................................................................... 115
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2.37.4 Results sheet ......................................................................................................................................... 116 2.38 Test No. 01-0038SSLSB_FEM: Thin cylinder under a uniform radial pressure ............................................ 117
2.38.1 Description sheet ................................................................................................................................... 117 2.38.2 Overview ................................................................................................................................................ 117 2.38.3 Stresses in all points .............................................................................................................................. 118 2.38.4 Cylinder deformation in all points ........................................................................................................... 118 2.38.5 Results sheet ......................................................................................................................................... 118
2.39 Test No. 01-0039SSLSB_FEM: Square plate under planar stresses.............................................................. 119 2.39.1 Description sheet ................................................................................................................................... 119 2.39.2 Overview ................................................................................................................................................ 119 2.39.3 Displacement of the model in the linear elastic range ............................................................................ 120 2.39.4 Results sheet ......................................................................................................................................... 121
2.40 Test No. 01-0040SSLSB_FEM: Stiffen membrane ........................................................................................... 122 2.40.1 Description sheet ................................................................................................................................... 122 2.40.2 Overview ................................................................................................................................................ 122 2.40.3 Results of the model in the linear elastic range...................................................................................... 123 2.40.4 Results sheet ......................................................................................................................................... 124
2.41 Test No. 01-0041SSLLB_FEM: Beam on two supports considering the shear force ................................... 125 2.41.1 Description sheet ................................................................................................................................... 125 2.41.2 Overview ................................................................................................................................................ 125 2.41.3 Vertical displacement of the model in the linear elastic range................................................................ 126 2.41.4 Results sheet ......................................................................................................................................... 126
2.42 Test No. 01-0042SSLSB_FEM: Thin cylinder under a uniform axial load...................................................... 127 2.42.1 Description sheet ................................................................................................................................... 127 2.42.2 Overview ................................................................................................................................................ 127 2.42.3 Stress in all points .................................................................................................................................. 128 2.42.4 Cylinder deformation at the free end ...................................................................................................... 128 2.42.5 Results sheet ......................................................................................................................................... 129
2.43 Test No. 01-0043SSLSB_FEM: Thin cylinder under a hydrostatic pressure................................................. 130 2.43.1 Description sheet ................................................................................................................................... 130 2.43.2 Overview ................................................................................................................................................ 130 2.43.3 Stresses ................................................................................................................................................. 131 2.43.4 Cylinder deformation .............................................................................................................................. 131 2.43.5 Results sheet ......................................................................................................................................... 132
2.44 Test No. 01-0044SSLSB_FEM: Thin cylinder under its self weight................................................................ 133 2.44.1 Description sheet ................................................................................................................................... 133 2.44.2 Overview ................................................................................................................................................ 133 2.44.3 Stresses ................................................................................................................................................. 134 2.44.4 Cylinder deformation .............................................................................................................................. 134 2.44.5 Results sheet ......................................................................................................................................... 134
2.45 Test No. 01-0045SSLSB_FEM: Torus with uniform internal pressure ........................................................... 135 2.45.1 Description sheet ................................................................................................................................... 135 2.45.2 Overview ................................................................................................................................................ 135 2.45.3 Stresses ................................................................................................................................................. 136 2.45.4 Cylinder deformation .............................................................................................................................. 136 2.45.5 Results sheet ......................................................................................................................................... 136
2.46 Test No. 01-0046SSLSB_FEM: Spherical shell under internal pressure ....................................................... 137 2.46.1 Description sheet ................................................................................................................................... 137 2.46.2 Overview ................................................................................................................................................ 137 2.46.3 Stresses ................................................................................................................................................. 138 2.46.4 Cylinder deformation .............................................................................................................................. 138 2.46.5 Results sheet ......................................................................................................................................... 139
2.47 Test No. 01-0047SSLSB_FEM: Spherical shell under its self weight............................................................. 140
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2.47.1 Description sheet ................................................................................................................................... 140 2.47.2 Overview................................................................................................................................................ 140 2.47.3 Stresses................................................................................................................................................. 141 2.47.4 Cylinder radial deformation .................................................................................................................... 141 2.47.5 Results sheet ......................................................................................................................................... 142
2.48 Test No. 01-0048SSLSB_FEM: Pinch cylindrical shell ................................................................................... 143 2.48.1 Description sheet ................................................................................................................................... 143 2.48.2 Overview................................................................................................................................................ 143 2.48.3 Vertical displacement at point A............................................................................................................. 144 2.48.4 Results sheet ......................................................................................................................................... 144
2.49 Test No. 01-0049SSLSB_FEM: Spherical shell with holes ............................................................................. 145 2.49.1 Description sheet ................................................................................................................................... 145 2.49.2 Overview................................................................................................................................................ 145 2.49.3 Horizontal displacement at point A ........................................................................................................ 146 2.49.4 Results sheet ......................................................................................................................................... 146
2.50 Test No. 01-0050SSLSB_FEM: Spherical dome under a uniform external pressure ................................... 147 2.50.1 Description sheet ................................................................................................................................... 147 2.50.2 Overview................................................................................................................................................ 147 2.50.3 Horizontal displacement and exterior meridian stress............................................................................ 148 2.50.4 Results sheet ......................................................................................................................................... 149
2.51 Test No. 01-0051SSLSB_FEM: Simply supported square plate under a uniform load ................................ 150 2.51.1 Description sheet ................................................................................................................................... 150 2.51.2 Overview................................................................................................................................................ 150 2.51.3 Vertical displacement and bending moment at the center of the plate................................................... 151 2.51.4 Results sheet ......................................................................................................................................... 151
2.52 Test No. 01-0052SSLSB_FEM: Simply supported rectangular plate under a uniform load......................... 152 2.52.1 Description sheet ................................................................................................................................... 152 2.52.2 Overview................................................................................................................................................ 152 2.52.3 Vertical displacement and bending moment at the center of the plate................................................... 153 2.52.4 Results sheet ......................................................................................................................................... 153
2.53 Test No. 01-0053SSLSB_FEM: Simply supported rectangular plate under a uniform load......................... 154 2.53.1 Description sheet ................................................................................................................................... 154 2.53.2 Overview................................................................................................................................................ 154 2.53.3 Vertical displacement and bending moment at the center of the plate................................................... 155 2.53.4 Results sheet ......................................................................................................................................... 155
2.54 Test No. 01-0054SSLSB_FEM: Simply supported rectangular plate loaded with punctual force and moments............................................................................................................................................................. 156 2.54.1 Description sheet ................................................................................................................................... 156 2.54.2 Overview................................................................................................................................................ 156 2.54.3 Vertical displacement at C ..................................................................................................................... 157 2.54.4 Results sheet ......................................................................................................................................... 157
2.55 Test No. 01-0055SSLSB_FEM: Shear plate perpendicular to the medium surface ...................................... 158 2.55.1 Description sheet ................................................................................................................................... 158 2.55.2 Overview................................................................................................................................................ 158 2.55.3 Vertical displacement at C ..................................................................................................................... 159 2.55.4 Results sheet ......................................................................................................................................... 159
2.56 Test No. 01-0056SSLLB_FEM: Triangulated system with hinged bars ......................................................... 160 2.56.1 Description sheet ................................................................................................................................... 160 2.56.2 Overview................................................................................................................................................ 160 2.56.3 Tension force in BD bar ......................................................................................................................... 161 2.56.4 Vertical displacement at D ..................................................................................................................... 161 2.56.5 Results sheet ......................................................................................................................................... 161
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2.57 Test No. 01-0057SSLSB_FEM: 0.01m thick plate fixed on its perimeter, loaded with a uniform pressure .............................................................................................................................................................. 162 2.57.1 Description sheet ................................................................................................................................... 162 2.57.2 Overview ................................................................................................................................................ 162 2.57.3 Vertical displacement at C...................................................................................................................... 163 2.57.4 Results sheet ......................................................................................................................................... 163
2.58 Test No. 01-0058SSLSB_FEM: 0.01333 m thick plate fixed on its perimeter, loaded with a uniform pressure .............................................................................................................................................................. 164 2.58.1 Description sheet ................................................................................................................................... 164 2.58.2 Overview ................................................................................................................................................ 164 2.58.3 Vertical displacement at C...................................................................................................................... 165 2.58.4 Results sheet ......................................................................................................................................... 165
2.59 Test No. 01-0059SSLSB_FEM: 0.02 m thick plate fixed on its perimeter, loaded with a uniform pressure ........ 166 2.59.1 Description sheet ................................................................................................................................... 166 2.59.2 Overview ................................................................................................................................................ 166 2.59.3 Vertical displacement at C...................................................................................................................... 167 2.59.4 Results sheet ......................................................................................................................................... 167
2.60 Test No. 01-0060SSLSB_FEM: 0.05 m thick plate fixed on its perimeter, loaded with a uniform pressure .............................................................................................................................................................. 168 2.60.1 Description sheet ................................................................................................................................... 168 2.60.2 Overview ................................................................................................................................................ 168 2.60.3 Vertical displacement at C...................................................................................................................... 169 2.60.4 Results sheet ......................................................................................................................................... 169
2.61 Test No. 01-0061SSLSB_FEM: 0.1 m thick plate fixed on its perimeter, loaded with a uniform pressure .............................................................................................................................................................. 170 2.61.1 Description sheet ................................................................................................................................... 170 2.61.2 Overview ................................................................................................................................................ 170 2.61.3 Vertical displacement at C...................................................................................................................... 171 2.61.4 Results sheet ......................................................................................................................................... 171
2.62 Test No. 01-0062SSLSB_FEM: 0.01 m thick plate fixed on its perimeter, loaded with a punctual force .... 172 2.62.1 Description sheet ................................................................................................................................... 172 2.62.2 Overview ................................................................................................................................................ 172 2.62.3 Vertical displacement at point C (center of the plate) ............................................................................. 173 2.62.4 Results sheet ......................................................................................................................................... 173
2.63 Test No. 01-0063SSLSB_FEM: 0.01333 m thick plate fixed on its perimeter, loaded with a punctual force ....... 174 2.63.1 Description sheet ................................................................................................................................... 174 2.63.2 Overview ................................................................................................................................................ 174 2.63.3 Vertical displacement at point C (the center of the plate) ....................................................................... 175 2.63.4 Results sheet ......................................................................................................................................... 175
2.64 Test No. 01-0064SSLSB_FEM: 0.02 m thick plate fixed on its perimeter, loaded with a punctual force.................................................................................................................................................................... 176 2.64.1 Description sheet ................................................................................................................................... 176 2.64.2 Overview ................................................................................................................................................ 176 2.64.3 Vertical displacement at point C (the center of the plate) ....................................................................... 177 2.64.4 Results sheet ......................................................................................................................................... 177
2.65 Test No. 01-0065SSLSB_FEM: 0.05 m thick plate fixed on its perimeter, loaded with a punctual force .... 178 2.65.1 Description sheet ................................................................................................................................... 178 2.65.2 Overview ................................................................................................................................................ 178 2.65.3 Vertical displacement at point C center of the plate) .............................................................................. 179 2.65.4 Results sheet ......................................................................................................................................... 179
2.66 Test No. 01-0066SSLSB_FEM: 0.1 m thick plate fixed on its perimeter, loaded with a punctual force ...... 180 2.66.1 Description sheet ................................................................................................................................... 180 2.66.2 Overview ................................................................................................................................................ 180 2.66.3 Vertical displacement at point C (center of the plate) ............................................................................. 181
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2.66.4 Results sheet ......................................................................................................................................... 181 2.67 Test No. 01-0067SDLLB_FEM: Vibration mode of a thin piping elbow in space (case 1)............................ 182
2.67.1 Description sheet ................................................................................................................................... 182 2.67.2 Overview................................................................................................................................................ 182 2.67.3 Eigen modes frequencies ...................................................................................................................... 183 2.67.4 Results sheet ......................................................................................................................................... 183
2.68 Test No. 01-0068SDLLB_FEM: Vibration mode of a thin piping elbow in space (case 2)............................ 184 2.68.1 Description sheet ................................................................................................................................... 184 2.68.2 Overview................................................................................................................................................ 184 2.68.3 Eigen modes frequencies ...................................................................................................................... 185 2.68.4 Results sheet ......................................................................................................................................... 185
2.69 Test No. 01-0069SDLLB_FEM: Vibration mode of a thin piping elbow in space (case 3)............................ 186 2.69.1 Description sheet ................................................................................................................................... 186 2.69.2 Overview................................................................................................................................................ 186 2.69.3 Eigen modes frequencies ...................................................................................................................... 187 2.69.4 Results sheet ......................................................................................................................................... 187
2.70 Test No. 01-0077SSLPB_FEM: Reactions on supports and bending moments on a 2D portal frame (Rafters) .............................................................................................................................................................. 188 2.70.1 Description sheet ................................................................................................................................... 188 2.70.2 Overview................................................................................................................................................ 188 2.70.3 Moments and actions on supports M.R. calculation on a 2D portal frame. ............................................ 189 2.70.4 Results sheet ......................................................................................................................................... 189
2.71 Test No. 01-0078SSLPB_FEM: Reactions on supports and bending moments on a 2D portal frame (Columns) ........................................................................................................................................................... 190 2.71.1 Description sheet ................................................................................................................................... 190 2.71.2 Overview................................................................................................................................................ 190 2.71.3 Moments and reactions on supports M.R. calculation on a 2D portal frame. ......................................... 191 2.71.4 Results sheet ......................................................................................................................................... 191
2.72 Test No. 01-0084SSLLB_FEM: Short beam on two hinged supports ............................................................ 192 2.72.1 Description sheet ................................................................................................................................... 192 2.72.2 Overview................................................................................................................................................ 192 2.72.3 Reference results................................................................................................................................... 192 2.72.4 Results sheet ......................................................................................................................................... 193
2.73 Test No. 01-0085SDLLB_FEM: Slender beam of variable rectangular section with fixed-free ends (β=5) .................................................................................................................................................................... 194 2.73.1 Description sheet ................................................................................................................................... 194 2.73.2 Overview................................................................................................................................................ 194 2.73.3 Reference results................................................................................................................................... 195 2.73.4 Results sheet ......................................................................................................................................... 197
2.74 Test No. 01-0086SDLLB_FEM: Slender beam of variable rectangular section (fixed-fixed)........................ 198 2.74.1 Description sheet ................................................................................................................................... 198 2.74.2 Overview................................................................................................................................................ 198 2.74.3 Reference results................................................................................................................................... 198 2.74.4 Results sheet ......................................................................................................................................... 199
2.75 Test No. 01-0089SSLLB_FEM: Plane portal frame with hinged supports..................................................... 200 2.75.1 Description sheet ................................................................................................................................... 200 2.75.2 Overview................................................................................................................................................ 200 2.75.3 Calculation method used to obtain the reference solution ..................................................................... 201 2.75.4 Reference values................................................................................................................................... 201 2.75.5 Results sheet ......................................................................................................................................... 201
2.76 Test No. 01-0090HFLSB_FEM: Simply supported beam in Eulerian buckling with a thermal load ............ 202 2.76.1 Description sheet ................................................................................................................................... 202 2.76.2 Overview................................................................................................................................................ 202 2.76.3 Displacement of the model in the linear elastic range............................................................................ 203
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2.76.4 Results sheet ......................................................................................................................................... 204 2.77 Test No. 01-0091HFLLB_FEM: Double fixed beam in Eulerian buckling with a thermal load...................... 205
2.77.1 Description sheet ................................................................................................................................... 205 2.77.2 Overview ................................................................................................................................................ 205 2.77.3 Displacement of the model in the linear elastic range ............................................................................ 206 2.77.4 Results sheet ......................................................................................................................................... 206
2.78 Test No. 01-0092HFLLB_FEM: Cantilever beam in Eulerian buckling with thermal load............................. 207 2.78.1 Description sheet ................................................................................................................................... 207 2.78.2 Overview ................................................................................................................................................ 207 2.78.3 Displacement of the model in the linear elastic range ............................................................................ 207 2.78.4 Results sheet ......................................................................................................................................... 208
2.79 Test No. 01-0094SSLLB_FEM: 3D bar structure with elastic support ........................................................... 209 2.79.1 Description sheet ................................................................................................................................... 209 2.79.2 Overview ................................................................................................................................................ 209 2.79.3 Results ................................................................................................................................................... 210 2.79.4 Results sheet ......................................................................................................................................... 214
2.80 Test No. 01-0095SDLLB_FEM: Fixed/free slender beam with centered mass .............................................. 215 2.80.1 Description sheet ................................................................................................................................... 215 2.80.2 Test data ................................................................................................................................................ 215 2.80.3 Reference results ................................................................................................................................... 216 2.80.4 Results sheet ......................................................................................................................................... 219
2.81 Test No. 01-0096SDLLB_FEM: Fixed/free slender beam with eccentric mass or inertia ............................. 220 2.81.1 Description sheet ................................................................................................................................... 220 2.81.2 Problem data.......................................................................................................................................... 220 2.81.3 Reference frequencies ........................................................................................................................... 221 2.81.4 Results sheet ......................................................................................................................................... 222
2.82 Test No. 01-0097SDLLB_FEM: Double cross with hinged ends..................................................................... 223 2.82.1 Description sheet ................................................................................................................................... 223 2.82.2 Problem data.......................................................................................................................................... 223 2.82.3 Reference frequencies ........................................................................................................................... 224 2.82.4 Results sheet ......................................................................................................................................... 225
2.83 Test No. 01-0098SDLLB_FEM: Simple supported beam in free vibration ..................................................... 226 2.83.1 Description sheet ................................................................................................................................... 226 2.83.2 Problem data.......................................................................................................................................... 226 2.83.3 Reference frequencies ........................................................................................................................... 227 2.83.4 Results sheet ......................................................................................................................................... 228
2.84 Test No. 01-0099HSLSB_FEM: Membrane with hot point............................................................................... 229 2.84.1 Description sheet ................................................................................................................................... 229 2.84.2 Problem data.......................................................................................................................................... 229 2.84.3 σyy stress at point A: ............................................................................................................................... 231 2.84.4 Results sheet ......................................................................................................................................... 231
2.85 Test No. 01-0100SSNLB_FEM: Beam on 3 supports with T/C (k = 0)............................................................. 232 2.85.1 Description sheet ................................................................................................................................... 232 2.85.2 Overview ................................................................................................................................................ 232 2.85.3 References solutions.............................................................................................................................. 233 2.85.4 Results sheet ......................................................................................................................................... 234
2.86 Test No. 01-0101SSNLB_FEM: Beam on 3 supports with T/C (k → ∞)........................................................... 235 2.86.1 Description sheet ................................................................................................................................... 235 2.86.2 Overview ................................................................................................................................................ 235 2.86.3 References solutions.............................................................................................................................. 236 2.86.4 Results sheet ......................................................................................................................................... 237
2.87 Test No. 01-0102SSNLB_FEM: Beam on 3 supports with T/C (k = -10000 N/m)............................................ 238 2.87.1 Description sheet ................................................................................................................................... 238
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2.87.2 Overview................................................................................................................................................ 238 2.87.3 References solutions ............................................................................................................................. 239 2.87.4 Results sheet ......................................................................................................................................... 240
2.88 Test No. 01-0103SSLLB_FEM: Linear system of truss beams....................................................................... 241 2.88.1 Description sheet ................................................................................................................................... 241 2.88.2 Overview................................................................................................................................................ 241 2.88.3 References solutions ............................................................................................................................. 242 2.88.4 Results sheet ......................................................................................................................................... 243
2.89 Test No. 01-0104SSNLB_FEM: Non linear system of truss beams................................................................ 244 2.89.1 Description sheet ................................................................................................................................... 244 2.89.2 Overview................................................................................................................................................ 244 2.89.3 References solutions ............................................................................................................................. 245 2.89.4 Results sheet ......................................................................................................................................... 246
2.90 Test No. 02-0112SMLLB_P92: Study of a mast subjected to an earthquake................................................ 247 2.90.1 Description sheet ................................................................................................................................... 247 2.90.2 Model overview...................................................................................................................................... 247 2.90.3 Material strength model ......................................................................................................................... 247 2.90.4 Seismic hypothesis in conformity with PS92 regulation ......................................................................... 248 2.90.5 Modal analysis ....................................................................................................................................... 248 2.90.6 Spectral study ........................................................................................................................................ 249 2.90.7 Results sheet ......................................................................................................................................... 251
2.91 Test No. 02-0158SSLLB_B91: Linear element in combined bending/tension - without compressed reinforcements - Partially tensioned section................................................................................................... 252 2.91.1 Description sheet ................................................................................................................................... 252 2.91.2 Overview................................................................................................................................................ 252 2.91.3 Reinforcement calculation...................................................................................................................... 253 2.91.4 Results sheet ......................................................................................................................................... 255
2.92 Test No. 02-0162SSLLB_B91: Linear element in simple bending - without compressed reinforcement... 256 2.92.1 Description sheet ................................................................................................................................... 256 2.92.2 Overview................................................................................................................................................ 256 2.92.3 Reinforcement calculation...................................................................................................................... 257 2.92.4 Results sheet ......................................................................................................................................... 258
2.93 Test No. 03-0206SSLLG_CM66: Design of a Steel Structure according to CM66. ....................................... 259 2.93.1 Data ....................................................................................................................................................... 259 2.93.2 Effel Structure results............................................................................................................................. 261 2.93.3 CM66 Effel Expertise results.................................................................................................................. 262 2.93.4 Effel Structure / Advance Design comparison........................................................................................ 267
2.94 Test No. 03-0207SSLLG_CM66: Design of a 2D portal frame......................................................................... 269 2.94.1 Data ....................................................................................................................................................... 269 2.94.2 Effel Structure Results ........................................................................................................................... 270 2.94.3 Effel Expert CM results .......................................................................................................................... 271 2.94.4 Effel Structure / Advance Design comparison........................................................................................ 274
2.95 Test No. 03-0208SSLLG_BAEL91: Design of a concrete floor with an opening........................................... 276 2.95.1 Data ....................................................................................................................................................... 276 2.95.2 Effel Structure Results ........................................................................................................................... 279 2.95.3 Effel RC Expert Results ......................................................................................................................... 281 2.95.4 Effel Structure / Advance Design comparison........................................................................................ 283
3 EUROCODES 1 TESTS DESCRIPTION........................................................................................................................... 285 Wind calculation .............................................................................................................................................................. 286 3.1 Portal frame with 11.31° angle – Example A.fto .............................................................................................. 286
3.1.1 Wind pressure calculation...................................................................................................................... 286 3.1.2 Cpe and Wind force calculation for front wind (Y).................................................................................. 294
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3.1.3 Cpe and Wind force calculation for parallel wind (X) .............................................................................. 306 3.2 Monopitch frame with 15 ° angle – Example B.fto........................................................................................... 313
3.2.1 Lateral Wind (X direction): Cpe calculation on rooftop ........................................................................... 313 3.3 Portal frame with 10° angle – Example C.fto.................................................................................................... 316
3.3.1 Lateral Wind (X): Cpe calculation and Wind force calculation ................................................................ 316 Snow calculation .............................................................................................................................................................. 318 3.4 Portal frame with 11.31° angle – Example A.fto............................................................................................... 318
4 EUROCODES 2 TESTS DESCRIPTION ........................................................................................................................... 321 4.1 Test No. 01 – EC2: Minimum reinforcement of a beam................................................................................... 322
4.1.1 Overview ................................................................................................................................................ 322 4.1.2 Reference results ................................................................................................................................... 323 4.1.3 Results sheet ......................................................................................................................................... 323
4.2 Test No. 02 – EC2: Longitudinal reinforcement of a beam under a linear load - horizontal level behavior law ....................................................................................................................................................... 324 4.2.1 Overview ................................................................................................................................................ 324 4.2.2 Reference results ................................................................................................................................... 325 4.2.3 Results sheet ......................................................................................................................................... 325
4.3 Test No. 03 – EC2: Longitudinal reinforcement of a beam under a linear load - inclined stress strain behavior law ....................................................................................................................................................... 326 4.3.1 Overview ................................................................................................................................................ 326 4.3.2 Reference results ................................................................................................................................... 327 4.3.3 Results sheet ......................................................................................................................................... 328
4.4 Test No. 04 – EC2: Longitudinal reinforcement of a beam under a concentrated load - horizontal level behavior law ....................................................................................................................................................... 329 4.4.1 Overview ................................................................................................................................................ 329 4.4.2 Reference results ................................................................................................................................... 330 4.4.3 Results sheet ......................................................................................................................................... 330
4.5 Test No. 05 – EC2: Longitudinal reinforcement of a beam under a linear load - horizontal level behavior law ....................................................................................................................................................... 331 4.5.1 Overview ................................................................................................................................................ 331 4.5.2 Reference results ................................................................................................................................... 332 4.5.3 Results sheet ......................................................................................................................................... 332
4.6 Test No. 06 – EC2: Transverse reinforcement of a beam subjected to a linear load.................................... 333 4.6.1 Overview ................................................................................................................................................ 333 4.6.2 Reference results ................................................................................................................................... 334 4.6.3 Results sheet ......................................................................................................................................... 335
4.7 Test No. 07 – EC2: Longitudinal reinforcement of a beam under a linear load - horizontal level behavior law ....................................................................................................................................................... 336 4.7.1 Overview ................................................................................................................................................ 336 4.7.2 Reference results ................................................................................................................................... 337 4.7.3 Results sheet ......................................................................................................................................... 338
5 EUROCODES 3 TESTS DESCRIPTION ........................................................................................................................... 339 5.1 Test No. 01 – EC3: 2D frame Design................................................................................................................. 340
5.1.1 Data........................................................................................................................................................ 340 5.1.2 Axial forces – N ...................................................................................................................................... 342 5.1.3 Shear forces – T..................................................................................................................................... 342 5.1.4 Bending moments – M ........................................................................................................................... 342 5.1.5 Classification of the beam cross section (IPE 240) ................................................................................ 343 5.1.6 Resistance of the beam cross section (IPE 240).................................................................................... 344 5.1.7 Classification of the column cross section (HEA 200) ............................................................................ 351 5.1.8 Resistance of the column cross section (HEA 200) ...............................................................................352
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5.2 Example 1 - Class of cross section (EC3)........................................................................................................ 359 5.2.1 Overview................................................................................................................................................ 359
5.3 Example 2 - Class of cross section (EC3)........................................................................................................ 362 5.3.1 Overview................................................................................................................................................ 362
5.4 Example 3 - Class of cross section (EC3)........................................................................................................ 365 5.4.1 Overview................................................................................................................................................ 365
5.5 Example 4: Tension column – design value of the resistance (EC3) ............................................................ 367 5.6 Example 5: Tension column – design value of the resistance (EC3) ............................................................ 368
1 Introduction
Before official releases, each version of GRAITEC software, includingAdvance Design, undergoes a series of validation tests from the"standard tests".
This validation is performed in parallel and in addition to the Beta-Test in order to obtain the "operational version" status.
At the moment, the database for automatic test includes 96 testswhich are codified and stored in a precise manner.
Each test is the subject of several reference documents:
Test description document.
Result sheet.
Information sheet corresponding to the tested model.
The test description documents coding is detailed below.
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1.1 Test description documents coding
The test description documents coding is summarized in the following table:
Year Regulations
Test number
Application field Analysis type
Behavior type Model type Results comparison
O1 OOO1 Static Linear Linear O2 OOO2
Structure mechanics S L L
To theoretical reference
Finite elements
. OOO3 S Dynamic B FEM
. . D Non-linear Planar Eurocodes X
. . N S
To standard mesh
ECX . . Thermomechanical
Eulerian buckling N BAEL
H F Plane (2D) B91 Spectral P
To a different software CM66
M A C66
Transitory Discrete Blind CB71 Thermal T D G C71
T Stationary . P .
The first column of the table corresponds to the last two digits of the year of creation of the test.
The second column of the table contains the number of the test (4 digits). The first test is numbered 0001. (9999 tests can be created each year.)
Columns 3 to 6 of the table are from the Structure Calculation Software Validation Guide of AFNOR. They regard all GRAITEC software according to their application fields.
Regarding the selection of the analysis type, if two types of analysis are used for the same test, the test of the highest level is accepted for the codification (the test from the lowest level in the corresponding column); for example: a test defined in static and in dynamic will get the dynamic corresponding short name (D)
Regarding the choice of model type, the surface denomination includes models consisting of planar elements only, as well as models consisting of planar and linear elements.
The results comparison column presents the reference taken into account to validate the results obtained with GRAITEC software.
The last column contains the regulation used for the test:
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Type Regulations Used code
Climatic NV65-84 N65
DIN-1055 D55 NBE-EA-95-EHE NBE NBN-B03 NBN RSA-98 R98 STAS 10101/21-92/20-90 S01
Seismic NCSE94 N94
P100-92 P00 PS69 P69 PS92 P92 RPA88 R88 RPA99 R99 RSA-98 S98 SI413 S13
Reinforced concrete ACI ACI
BAEL B91 DIN DIN EC2 EC2 EHE EHE STAS 10107/0-90 S07
Steelwork CM66 C66
EA-95 E95 EC3 EC3
Timber construction CB71 C71
Finite elements FEM FEM
The test description documents coding consists of the 8 columns. For the results comparison documents, the version number used is added next (without dot).
1.2 Test description documents coding example
SSLL: Analysis FEM: Regulation
01: Date 01-0001SSLLA_FEM
0001: Test number A: Results comparison
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1.3 Results comparison documents coding example
01-0001SSLLA_FEM-092N
92N: Version used for the test calculation
01-0001SSLLA_FEM-101M 101M: Version used for the test calculation
1.4 Margin of error
The acceptable margin of error for a test validation is:
Static 2% Dynamic 5%
Eulerian buckling 5% Spectral 5%
Stationary 5% Transitory 5% Climatic 5% Seismic 5%
Reinforced concrete 10% Steelwork 10%
Timber construction 10%
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1.5 Synthetic sheet of 2010 version
Software Solver Code Title Reference value Calculated value Deviation
Advance Design 2010 CM2 01-0001SSLSB_FEM Cantilever Rectangular plate -9.71 -9.59 -1.24%
Advance Design 2010 CM2 01-0002SSLLB_FEM System of two bars with three hinges -0.30 -0.30 0.00%
Advance Design 2010 CM2 01-0003SSLSB_FEM Circular plate under uniform load -6.50 -6.47 -0.46%
Advance Design 2010 CM2 01-0004SDLLB_FEM Slender beam with variable section (fixed-free) 1112.28 1075.70 -3.29%
Advance Design 2010 CM2 01-0005SSLLB_FEM Tied (sub-tensioned) beam 584584 584584 0.00%
Advance Design 2010 CM2 01-0006SDLLB_FEM Thin circular ring fixed in two points. 2801.5 2777.43 -0.86%
Advance Design 2010 CM2 01-0007SDLSB_FEM Thin lozenge-shaped plate fixed on one side (a = 0°) 8.7266 8.67 -0.65%
Advance Design 2010 CM2 01-0008SDLSB_FEM Thin lozenge-shaped plate fixed on one side (a = 15°) 22.1714 21.69 -2.17%
Advance Design 2010 CM2 01-0009SDLSB_FEM Thin lozenge-shaped plate fixed on one side (a = 30°) 25.4651 23.44 -7.95%
Advance Design 2010 CM2 01-0010SDLSB_FEM Thin lozenge-shaped plate fixed on one side (a = 45°) 26.3897 28.08 6.41%
Advance Design 2010 CM2 01-0011SDLLB_FEM Vibration mode of a thin piping elbow in plane (case 1) 119 120.09 0.92%
Advance Design 2010 CM2 01-0012SDLLB_FEM Vibration mode of a thin piping elbow in plane (case 2) 180 184.68 2.60%
Advance Design 2010 CM2 01-0013SDLLB_FEM Vibration mode of a thin piping elbow in plane (case 3) 25.300 24.961 -1.34%
Advance Design 2010 CM2 01-0014SDLLB_FEM Thin circular ring suspended by an elastic leg. 682.00 683.9 0.28%
Advance Design 2010 CM2 01-0015SSLLB_FEM Double fixed beam with a spring in the middle -0.11905 -0.11905 0.00%
Advance Design 2010 CM2 01-0014SDLLB_FEM Double fixed beam 26.228 25.758 -1.79%
Advance Design 2010 CM2 01-0017SDLLB_FEM Short beam on simple supports (on the neutral axis) 1498.295 1537.156 2.59%
Advance Design 2010 CM2 01-0018SDLLB_FEM Short beam on simple supports (eccentric) 902.2 945.35 4.78%
Advance Design 2010 CM2 01-0019SDLSB_FEM Thin square plate fixed on one side 136.0471 134.655 -1.02%
Advance Design 2010 CM2 01-0020SDLSB_FEM Thin rectangular plate simply supported on the perimeter 197.32 195.545 -0.90%
Advance Design 2010 CM2 01-0021SFLLB_FEM Cantilever beam in Eulerian buckling -98696 -98699.278 0.00%
Advance Design 2010 CM2 01-0022SDLSB_FEM Thin ring plate fixed on a hub 609.7 593.83 -2.60%
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Advance Design 2010 CM2 01-0023SDLLB_FEM Bending effects of a symmetrical portal frame 206 206.2 0.10%
Advance Design 2010 CM2 01-0024SSLLB_FEM Slender beam on two fixed supports -540 -540 0.00%
Advance Design 2010 CM2 01-0025SSLLB_FEM Slender beam on three supports 63000 63000 0.00%
Advance Design 2010 CM2 01-0026SSLLB_FEM Bimetallic: Fixed beams connected to a stiff element -0.125 -0.125 0.00%
Advance Design 2010 CM2 01-0027SSLLB_FEM Fixed thin arc in planar bending 0.3791 0.37891 -0.05%
Advance Design 2010 CM2 01-0028SSLLB_FEM Fixed thin arc in out of plane bending -96.5925 -96.5779 -0.02%
Advance Design 2010 CM2 01-0029SSLLB_FEM Double hinged thin arc in planar bending 5.3912 5.386 -0.10%
Advance Design 2010 CM2 01-0030SSLLB_FEM Portal frame with lateral connections 113.559 113.7044 0.13%
Advance Design 2010 CM2 01-0031SSLLB_FEM Truss with hinged bars under a punctual load 0.08839 0.08817 -0.25%
Advance Design 2010 CM2 01-0032SSLLB_FEM Beam on elastic soil, free ends 5759 5772.05 0.23%
Advance Design 2010 CM2 01-0033SFLLA_FEM EDF Pylon 2.77 2.830 2.17%
Advance Design 2010 CM2 01-0034SSLLB_FEM Beam on elastic soil, hinged ends 11674 11644.36 -0.25%
Advance Design 2010 CM2 01-0035SSLPB_FEM Plate with in plane bending and shear -80 -79.46 -0.68%
Advance Design 2010 CM2 01-0036SSLSB_FEM Simply supported square plate -0.158 -0.16491 4.37%
Advance Design 2010 CM2 01-0037SSLSB_FEM Caisson beam in torsion -0.11 -0.10 -9.09%
Advance Design 2010 CM2 01-0038SSLSB_FEM Thin cylinder under uniform radial pressure -2.38 x 10-6 -2.39 x 10-6 -0.42%
Advance Design 2010 CM2 01-0039SSLSB_FEM Square plate under planar stresses -14.66 -14.61 -0.34%
Advance Design 2010 CM2 01-0040SSLSB_FEM Stiffen membrane 11.55 11.55 0.00%
Advance Design 2010 CM2 01-0041SSLLB_FEM Beam on two supports considering the shear force -1.017 -1.017 0.00%
Advance Design 2010 CM2 01-0042SSLSB_FEM Thin cylinder under uniform axial load -7.14 x 10-7 -7,109 x 10-7 0.43%
Advance Design 2010 CM2 01-0043SSLSB_FEM Thin cylinder under a hydrostatic pressure -2.86 x 10-6 -2,854 x 10-6 -0.20%
Advance Design 2010 CM2 01-0044SSLSB_FEM Thin cylinder under its self weight 3.14 x 105 3.11 x 105 -0.95%
Advance Design 2010 CM2 01-0045SSLSB_FEM Torus with a uniform internal pressure 7.5 x 105 7.43 x 105 -0.94%
Advance Design 2010 CM2 01-0046SSLSB_FEM Spherical shell under internal pressure 8.33 x 10-7 8.34 x 10-7 0.12%
Advance Design 2010 CM2 01-0047SSLSB_FEM Spherical shell under its self weight -7.85 x 104 -7.90 x 104 -0.63%
Advance Design 2010 CM2 01-0048SSLSB_FEM Pinch cylindrical shell -113.9 x 10-3 -113.29 x 10-3 -0.53%
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Advance Design 2010 CM2 01-0049SSLSB_FEM Spherical shell with holes 94.0 92.13 -1.99%
Advance Design 2010 CM2 01-0050SSLSB_FEM Spherical dome under uniform external pressure -1.73 x 10-3 -2.14 x 10-3 23%
Advance Design 2010 CM2 01-0051SSLSB_FEM Simply supported square plate under a uniform load 0.0479 0.0471 -1.67%
Advance Design 2010 CM2 01-0052SSLSB_FEM Simply supported rectangular plate under a uniform load 1.106 x 10-2 1.102 x 10-2 -0.36%
Advance Design 2010 CM2 01-0053SSLSB_FEM Simply supported rectangular plate under a uniform load 1.416 x 10-2 1.401 x 10-2 -1.06%
Advance Design 2010 CM2 01-0054SSLSB_FEM Simply supported rectangular plate with punctual forces and moments -12.480 -12.667 1.50%
Advance Design 2010 CM2 01-0055SSLSB_FEM Shear plate perpendicular to the medium surface 35.37 x 10-3 35.67 x 10-3 0.85%
Advance Design 2010 CM2 01-0056SSLLB_FEM Triangulated system with hinged bars 43633 43688 0.13%
Advance Design 2010 CM2 01-0057SSLSB_FEM 0.01m thick plate fixed on its perimeter, loaded with a uniform pressure 0.66390 x 10-2 0.65879 x 10-2 -0.77%
Advance Design 2010 CM2 01-0058SSLSB_FEM 0.01333 m thick plate fixed on its perimeter, loaded with a uniform pressure 0.28053 x 10-2 0.28045 x 10-2 -0.03%
Advance Design 2010 CM2 01-0059SSLSB_FEM 0.02 m thick plate fixed on its perimeter, loaded with a uniform pressure 0.83480 x 10-2 0.82839 x 10-2 -0.76%
Advance Design 2010 CM2 01-0060SSLSB_FEM 0.05 m thick plate fixed on its perimeter, loaded with a uniform pressure 0.55474 x 10-3 0.55170 x 10-3 -0.55%
Advance Design 2010 CM2 01-0061SSLSB_FEM 0.1 m thick plate fixed on its perimeter, loaded with a uniform pressure 0.78661 x 10-4 0.78180 x 10-4 -0.61%
Advance Design 2010 CM2 01-0062SSLSB_FEM 0.01 m thick plate fixed on its perimeter, loaded with a punctual force 0.29579 0.29215 -1.23%
Advance Design 2010 CM2 01-0063SSLSB_FEM 0.01333 m thick plate fixed on its perimeter, loaded with a punctual force 0.11837 0.12458 5.25%
Advance Design 2010 CM2 01-0064SSLSB_FEM 0.02 m thick plate fixed on its perimeter, loaded with a punctual force 0.037454 0.036980 -1.27%
Advance Design 2010 CM2 01-0065SSLSB_FEM 0.05 m thick plate fixed on its perimeter, loaded with a punctual force 0.25946 x 10-2 0.25723 x 10-2 -0.86%
Advance Design 2010 CM2 01-0066SSLSB_FEM 0.1m thick plate fixed on its perimeter, loaded with a punctual force 0.42995 x 10-2 0.41209 x 10-2 -4.15%
Advance Design 2010 CM2 01-0067SDLLB_FEM Vibration mode of a thin piping elbow in space (case 1) 125 120.09 -3.93%
Advance Design 2010 CM2 01-0068SDLLB_FEM Vibration mode of a thin piping elbow in space (case 2) 100 94.62 -5.38%
Advance Design 2010 CM2 01-0069SDLLB_FEM Vibration mode of a thin piping elbow in space (case 3) 24.800 24.43 -1.49%
Advance Design 2010 CM2 01-0077SSLPB_FEM Reactions on supports and bending moments on a 2D portal frame (Rafters) 1671 1673.35 0.14%
Advance Design 2010 CM2 01-0078SSLPB_FEM Reactions on supports and bending moments on a 2D portal frame (Columns) -302.7 -302.06 -0.21%
Advance Design 2010 CM2 01-0084SSLLB_FEM Short beam on two hinged supports -1.25926 -1.25926 0.00%
Advance Design 2010 CM2 01-0085SDLLB_FEM Slender beam of variable rectangular section with fixed-free ends (beta = 5) 56.55 58.49 3.43%
Advance Design 2010 CM2 01-0086SDLLB_FEM Slender beam Slender beam of variable rectangular section fixed-fixed 143.303 145.88 1.80%
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Advance Design 2010 CM2 01-0089SSLLB_FEM 2D portal frame with hinged supports -0.03072 -0.03072 0.00%
Advance Design 2010 CM2 01-0090HFLSB-FEM Simply supported beam in Eulerian buckling with a thermal load -40774 -41051.95 0.68%
Advance Design 2010 CM2 01-0091HFLLB_FEM Double fixed beam in Eulerian buckling with thermal load -117.724 -118.08 0.30%
Advance Design 2010 CM2 01-0092HFLLB _FEM Cantilever beam in Eulerian buckling with thermal load -0.5 -0.5 0.00%
Advance Design 2010 CM2 01-0094SSLLB_FEM 3D bar structure with elastic support -1436 -1436,55 -3.83%
Advance Design 2010 CM2 01-0095SDLLB_FEM Fixed/free slender beam with a centered mass 16.07 16.06 -0.06%
Advance Design 2010 CM2 01-0096SDLLB_FEM Fixed/free slender beam with eccentric mass or inertia 61.61 63.091 2.40%
Advance Design 2010 CM2 01-0097SDLLB_FEM Double cross with hinged ends 57.39 56.06 -2.32%
Advance Design 2010 CM2 01-0098SDLLB_FEM Simply supported beam in free vibration 42.649 43.11 1.08%
Advance Design 2010 CM2 01-0099HSLSB_FEM Membrane with hot point 50 50.87 1.74%
Advance Design 2010 CM2 01-0100SSNLB_FEM Beam on 3 supports with T/C (k = 0) -0.000153 -0.000153 0.00%
Advance Design 2010 CM2 01-0101SSNLB_FEM Beam on 3 supports with T/C (k tends to infinite) -93.75 -93.649 -0.11%
Advance Design 2010 CM2 01-0102SSNLB_FEM Beam on 3 supports with T/C (k = -10000 N/m) -58.15 -58.117 -0.06%
Advance Design 2010 CM2 01-0103SSLLB_FEM Linear system of truss beams 0.000649 0.000649 0.00%
Advance Design 2010 CM2 01-0104SSNLB_FEM Non-linear system of truss beams 0.001195 0.001190 -0.42%
Advance Design 2010 CM2 02-0112SMLLB_P92 Study of a mast subjected to an earthquake 2.929 2.927 -0.07%
Advance Design 2010 CM2 02-158SSLLB_BAEL91 Linear element in combined bending/tension - without compressed reinforcements - partially tensioned section -- -- Ok
Advance Design 2010 CM2 02-162SSLLB_BAEL91 Linear element in simple bending - without compressed reinforcement -- -- Ok
Advance Design 2010 CM2 03-0206SSLLG_CM66 Optimization of a Steel Structure according to CM66 339.74 347.46 2.20%
Advance Design 2010 CM2 03-0207SSLLG_CM66 Design of a 2D portal frame 230.34 230.34 0.00%
Advance Design 2010 CM2 03-0208SSLLG_BAEL91 Design of a concrete floor with an opening 0.18 0.175 2.85%
2 Detailed tests description
Test 01-0001SSLSB_FEM
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2.1 Test No. 01-0001SSLSB_FEM: Cantilever rectangular plate
2.1.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 01/89.
Analysis type: linear static.
Element type: planar.
2.1.2 Overview The 1 m long plate is fixed at one end and has a "p" planar load.
Cantilever rectangular plate Scale =1/4 01-0001SSLSB_FEM
Units
S.I.
Geometry
Thickness: e = 0.005 m,
Length: l = 1 m,
Width: b = 0.1 m.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
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Boundary conditions
Outer: Fixed at end x = 0,
Inner: None.
Loadings
External: Uniform load p = -1700 Pa on the upper surface,
Internal: None.
2.1.3 Displacement of the model in the linear elastic range Reference solution
The reference displacement is calculated for the unsupported end located at x = 1m.
u = bl4p8EIz =
0.1 x 14 x 1700
8 x 2.1 x 1011 x 0.1 x 0.0053
12 = -9.71 cm
Finite elements modeling
Planar element: plate, imposed mesh,
1100 nodes,
990 surface quadrangles.
Deformed shape
Deformed cantilever rectangular plate Scale =1/4 01-0001SSLSB_FEM
2.1.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 Free extremity cm -9.71 -9.59 -1.24%
Test 01-0002SSLLB_FEM
12
2.2 Test No. 01-0002SSLLB_FEM: System of two bars with three hinges
2.2.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 09/89;
Analysis type: linear static;
Element type: linear.
2.2.2 Overview A punctual load, "F", is hanged from two connected hinged bars, also hinged at extremities.
System of two bars with three hinges Scale =1/33 0002SSLLB_FEM
4.500 m
30° 30°
4.500 m
AA BB
CC
FF
X
Y
Z X
Y
Z
Units
I. S.
Geometry
Bars angle relative to horizontal: θ = 30°,
Bars length: l = 4.5 m,
Bar section: A = 3 x 10-4 m2.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa.
Boundary conditions
Outer: Hinged in A and B,
Inner: Hinge on C
Test 01-0002SSLLB_FEM
13
Loading
External: Punctual load in C: F = -21 x 103 N.
Internal: None.
2.2.3 Displacement of the model in C
Reference solution
uc = -3 x 10-3 m
Finite elements modeling
Linear element: beam, imposed mesh,
21 nodes,
20 linear elements.
Displacement shape
System of two bars with three hinges Scale =1/33 Displacement in C 0002SSLLB_FEM
2.2.4 Bars stresses
Reference solutions
σAC bar = 70 MPa
σBC bar = 70 MPa
Finite elements modeling
Linear element: beam, imposed mesh,
21 nodes,
20 linear elements.
Test 01-0002SSLLB_FEM
14
2.2.5 Shape of the stress diagram System of two bars with three hinges Scale =1/34
Bars stresses 0002SSLLB_FEM
2.2.6 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In C cm -0.30 -0.30 0.00%
Results comparison: tensile stress
Solver Positioning Units Reference AD 2010 Deviation CM2 AC bar MPa 70 70 0.00% CM2 BC bar MPa 70 70 0.00%
Test 01-0003SSLSB_FEM
15
2.3 Test No. 01-0003SSLSB_FEM: Circular plate under uniform load
2.3.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 03/89;
Analysis type: linear static;
Element type: planar.
2.3.2 Overview A circular plate of 5 mm thickness and 2 m diameter with an uniform load perpendicular on the plan of the plate.
Circular plate under uniform load Scale =1/10 01-0003SSLSB_FEM
Units
I. S.
Geometry
Circular plate radius: r = 1m,
Circular plate thickness: h = 0.005 m.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Test 01-0003SSLSB_FEM
16
Boundary conditions
Outer: Plate fixed on the side (in all points of its perimeter), For the modeling, we consider only a quarter of the plate and we impose symmetry conditions on some nodes (see the following model; yz plane symmetry condition):translation restrained nodes along x and rotation restrained nodes along y and z: translation restrained nodes along x and rotation restrained nodes along y and z:
Inner: None.
Loading
External: Uniform loads perpendicular on the plate: pZ = -1000 Pa,
Internal: None.
2.3.3 Vertical displacement of the model at the center of the plate
Reference solution
Circular plates form:
u = pr4
64D = -1000 x 14
64 x 2404 = - 6.50 x 10-3 m
with the plate radius coefficient: D = Eh3
12(1-ν2) = 2.1 x 1011 x 0.0053
12(1-0.32)
D = 2404
Finite elements modeling
Planar element: plate, imposed mesh,
70 nodes,
58 planar elements. Circular plate under uniform load Scale =1.5
Meshing 01-0003SSLSB_FEM
Test 01-0003SSLSB_FEM
17
Deformed shape
Circular plate under uniform load Scale =1.5 Deformed 01-0003SSLSB_FEM
2.3.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 Plate center mm -6.50 -6.47 -0.46%
Test 01-0004SDLLB_FEM
18
2.4 Test No. 01-0004SDLLB_FEM: Slender beam with variable section (fixed-free)
2.4.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 09/89;
Analysis type: modal analysis;
Element type: linear.
2.4.2 Overview Find the first eigen mode frequencies for a beam of variable section, subjected to its own weight only.
Slender beam with variable section (fixed-free) Scale =1/4 01-0004SDLLB_FEM
Units
I. S.
Geometry
Beam length: l = 1 m,
Initial section (in A): o Height: h1 = 0.04 m, o Width: b1 = 0.04 m, o Section: A1 = 1.6 x 10-3 m2, o Flexure moment of inertia relative to z-axis: Iz1 = 2.1333 x 10-7 m4,
Test 01-0004SDLLB_FEM
19
Final section (in B): o Height: h2 = 0.01 m, o Width: b2 = 0.01 m, o Section: A2 = 10-4 m2, o Flexure moment of inertia relative to z-axis: Iz2 = 8.3333 x 10-10 m4.
Materials properties
Longitudinal elastic modulus: E = 2 x 1011 Pa,
Density: 7800 kg/m3.
Boundary conditions
Outer: Fixed in A,
Inner: None.
Loading
External: None,
Internal: None.
2.4.3 Eigen mode frequencies
Reference solutions
Precise calculation by numerical integration of the differential equation of beams bending (Euler-Bernoulli theories):
∂2
∂x2 (EIz ∂2v∂x2 ) = -ρA
∂2v∂x2 where Iz and A vary with the abscissa.
The result is: fi = 12π λi
h2l2
E12ρ
λ1 λ2 λ3 λ4 λ5 23.289 73.9 165.23 299.7 478.1
Finite elements modeling
Linear element: variable beam, imposed mesh,
31 nodes,
30 linear elements.
Test 01-0004SDLLB_FEM
20
Eigen mode shapes
2.4.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation CM2 Mode 1 Hz 54.18 54.01 -0.31% CM2 Mode 2 Hz 171.94 170.58 -0.79% CM2 Mode 3 Hz 384.4 378.87 -1.44% CM2 Mode 4 Hz 697.24 681.31 -2.28% CM2 Mode 5 Hz 1112.28 1075.70 -3.29%
Test 01-0005SSLLB_FEM
21
2.5 Test No. 01-0005SSLLB_FEM: Tied (sub-tensioned) beam
2.5.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 13/89;
Analysis type: static, thermoelastic (plane problem);
Element type: linear.
2.5.2 Overview Beam reinforced by a system of hinged bars with a uniform linear load, p.
Tied (sub-tensioned) beam Scale =1/37 01-0005SSLLB_FEM
Units
I. S.
Geometry
Length: o AD = FB = a = 2 m, o DF = CE = b = 4 m, o CD = EF = c = 0.6 m, o AC = EB = d = 2.088 m, o Total length: L = 8 m,
AD, DF, FB Beams: o Section: A = 0.01516 m2, o Shear area: Ar = A / 2.5, o Inertia moment: I = 2.174 x 10-4 m4,
Test 01-0005SSLLB_FEM
22
CE Bar: o Section: A1 = 4.5 x 10-3 m2,
AC, EB bar: o Section: A2 = 4.5 x 10-3 m2,
CD, EF bars: o Section: A3 = 3.48 x 10-3 m2.
Materials properties
Isotropic linear elastic material,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Shearing module: G = 0.4x E.
Boundary conditions
Outer: Hinged in A, support connection in B (blocked vertical translation),
Inner: Hinged at bar ends: AC, CD, EF, EB.
Loading
External: Uniform linear load p = -50000 N/ml,
Internal: Shortening of the CE tie of δ = 6.52 x 10-3 m (dilatation coefficient: αCE = 1 x 10-5 /°C and temperature variation ΔT = -163°C).
2.5.3 Compression force in CE bar
Reference solution
The solution is established by considering the deformation effects due to the shear force and normal force:
μ = 1 - 43 x
aL
k = AAr
= 2.5
t = IA
γ = (L/c)2 x (1+ (A/A1) x (b/L) + 2 x (A/A2) x (d/a)2 x (d/L) + 2 x (A/A3) (c/a)2 x (c/L)
τ = k x [(2Et2) / (GaL)]
ρ = μ + γ + τ
μ0 = 1 – (a/L)2 x (2 – a/L)
τ0 = 6k x (E/G) x (t/L)2 x (1 + b/L)
ρ0 = μ0 + τ0
NCE = - (1/12) x (pL2/c) x (ρ0 /ρ) + (EI/(Lc2)) x (δ/ρ) = 584584 N
Finite elements modeling
Linear element: without meshing,
AD, DF, FB: S beam (considering the shear force deformations),
AC, CD, EF, EB: bar,
CE: beam, 6 nodes.
Test 01-0005SSLLB_FEM
23
Force diagrams
Tied (sub-tensioned) beam Scale =1/31 Compression force in CE bar
2.5.4 Bending moment at point H
Reference solution
MH = - (1/8) x pL2 x [1- (2/3) x (ρ0/ρ)] – (EI/(Lc)) x (δ/p) = 49249.5 N
Finite elements modeling
Linear element: without meshing,
AD, DF, FB: S beam (considering the shear force deformations),
AC, CD, EF, EB: bar,
CE: beam, 6 nodes.
Test 01-0005SSLLB_FEM
24
Shape of the bending moment diagram
Tied (sub-tensioned) beam Scale =1/31 Mz bending moment
2.5.5 Vertical displacement at point D
Reference solution
The reference displacement vD provided by AFNOR is determined by averaging the results of several software with implemented finite elements method.
vD = -0.5428 x 10-3 m
Finite elements modeling
Linear element: without meshing, o AD, DF, FB: S beam (considering the shear force deformations), o AC, CD, EF, EB: bar, o CE: beam,
6 nodes.
Test 01-0005SSLLB_FEM
25
Deformed shape
Tied (sub-tensioned) beam Scale =1/31 Deformed
2.5.6 Results sheet
Results comparison: Tension force
Solver Positioning Units Reference AD 2010 Deviation CM2 CE Bar N 584584 584584 0.00%
Test 01-0006SDLLB_FEM
26
2.6 Test No. 01-0006SDLLB_FEM: Thin circular ring fixed in two points
2.6.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 12/89;
Analysis type: modal analysis, plane problem;
Element type: linear.
2.6.2 Overview Research of the first eigen modes frequencies for a circular ring fixed in two points, subjected to its own weight only.
Thin circular ring fixed in two points Scale =1/2 01-0006SDLLB_FEM
Units I. S.
Geometry
Average radius of curvature: OA = OB = R = 0.1 m,
Angular spacing between points A and B: 120° ;
Rectangular straight section: oThickness: h = 0.005 m, oWidth: b = 0.010 m, oSection: A = 5 x 10-5 m2, oFlexure moment of inertia relative to the vertical axis: I = 1.042 x 10-10 m4,
Point coordinates: o O (0 ;0),
o A (-0.05 3 ; -0.05),
o B (0.05 3 ; -0.05).
Test 01-0006SDLLB_FEM
27
Materials properties
Longitudinal elastic modulus: E = 7.2 x 1010 Pa Poisson's ratio: ν = 0.3, Density: ρ = 2700 kg/m3.
Boundary conditions
Outer: Fixed at A and B, Inner: None.
Loading
External: None, Internal: None.
2.6.3 Eigen mode frequencies Reference solutions The deformation of the fixed ring is calculated from the deformations of the free-free thin ring
Symmetrical mode: o u’i = i cos(iθ) o v’i = sin (iθ)
o θ’i = 1-i2R sin (iθ)
Antisymmetrical mode: o u’i = i sin(iθ) o v’i = -cos (iθ)
o θ’i = 1-i2R cos (iθ)
From Green’s method results:
fj = π21
λj ⋅2R
h 12E
ρ with a support angle of 120°.
i 1 2 3 4 Symmetrical mode 4.8497 14.7614 23.6157
Antisymmetrical mode 1.9832 9.3204 11.8490 21.5545
Finite elements modeling
Linear element: beam, without meshing, 32 nodes, 32 linear elements.
Eigen mode shapes
Test 01-0006SDLLB_FEM
28
2.6.4 Results sheet
Results comparison: eigen mode frequencies
Solver
Eigen mode type Eigen mode order j i
Units Reference AD 2010 Deviation
CM2 Mode 1 1 antisymmetric 1 Hz 235.3 236.32 0.43% CM2 Mode 2 2 symmetric 1 Hz 575.3 578.52 0.56% CM2 Mode 3 3 antisymmetric 2 Hz 1105.7 1112.54 0.62% CM2 Mode 4 4 antisymmetric 3 Hz 1405.6 1414.22 0.61% CM2 Mode 5 5 symmetric 2 Hz 1751.1 1760 0.51% CM2 Mode 6 6 antisymmetric 4 Hz 2557 2569.97 0.51% CM2 Mode 7 7 symmetric 3 Hz 2801.5 2777.43 -0.86%
Test 01-0007SDLSB_FEM
29
2.7 Test No. 01-0007SDLSB_FEM: Thin lozenge-shaped plate fixed on one side (α = 0 °)
2.7.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLS 02/89;
Analysis type: modal analysis;
Element type: planar.
2.7.2 Overview The 10 mm thick plate is fixed on one side and is with its self weight only.
Thin lozenge-shaped plate fixed on one side Scale =1/10 01-0007SDLSB_FEM
Units
I. S.
Geometry
Thickness: t = 0.01 m, Side: a = 1 m, α = 0° Points coordinates:
o A ( 0 ; 0 ; 0 ) o B ( a ; 0 ; 0 ) o C ( 0 ; a ; 0 ) o D ( a ; a ; 0 )
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa, Poisson's ratio: ν = 0.3, Density: ρ = 7800 kg/m3.
Test 01-0007SDLSB_FEM
30
Boundary conditions
Outer: AB side fixed,
Inner: None.
Loading
External: None,
Internal: None.
2.7.3 Eigen mode frequencies relative to the α angle
Reference solution M. V. Barton formula for a side "a" lozenge, leads to the frequencies:
fj = ⋅⋅π 2a21
λi2
)1(12Et
2
2
ν−ρ where i = 1,2, and λi
2 = g(α).
α = 0 λ1
2 3.492 λ2
2 8.525
M.V. Barton noted the sensitivity of the result relative to the mode and the α angle. He acknowledged that the λi values were determined with a limited development of an insufficient order, which led to consider a reference value that is based on an experimental result, verified by an average of seven software that use the finite elements calculation method.
Finite elements modeling
Planar element: plate, imposed mesh,
61 nodes,
900 surface quadrangles.
Eigen mode shapes
2.7.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation CM2 Mode 1 Hz 8.7266 8.67 -0.65% CM2 Mode 2 Hz 21.3042 21.21 -0.44%
Test 01-0008SDLSB_FEM
31
2.8 Test No. 01-0008SDLSB_FEM: Thin lozenge-shaped plate fixed on one side (α = 15 °)
2.8.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLS 02/89;
Analysis type: modal analysis;
Element type: planar.
2.8.2 Overview The 10 mm thick plate is fixed on one side and is loaded with its self weight only.
Thin lozenge-shaped plate fixed on one side Scale =1/10 01-0008SDLSB_FEM
Units I. S.
Geometry
Thickness: t = 0.01 m,
Side: a = 1 m,
α = 15°
Points coordinates: o A ( 0 ; 0 ; 0 ) o B ( a ; 0 ; 0 ) o C ( 0.259a ; 0.966a ; 0 ) o D ( 1.259a ; 0.966a ; 0 )
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Test 01-0006SDLLB_FEM
32
Boundary conditions
Outer: AB side fixed,
Inner: None.
Loading
External: None,
Internal: None.
2.8.3 Eigen modes frequencies function by α angle
Reference solution
M. V. Barton formula for a lozenge of side "a" leads to the frequencies:
fj = ⋅⋅π 2a21
λi2
)1(12Et
2
2
ν−ρ where i = 1,2, or λi
2 = g(α).
α = 15° λ1
2 3.601 λ2
2 8.872
M. V. Barton noted the sensitivity of the result relative to the mode and the α angle. He acknowledged that the λi values were determined with a limited development of an insufficient order, which led to consider a reference value that is based on an experimental result, verified by an average of seven software that use the finite elements calculation method.
Finite elements modeling
Planar element: plate, imposed mesh,
961 nodes,
900 surface quadrangles.
Eigen mode shapes
2.8.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation CM2 Mode 1 Hz 8.999 8.95 -0.54% CM2 Mode 2 Hz 22.1714 21.69 -2.17%
Test 01-0009SDLSB_FEM
33
2.9 Test No. 01-0009SDLSB_FEM: Thin lozenge-shaped plate fixed on one side (α = 30 °)
2.9.1 Description sheet Reference: Structure Calculation Software Validation Guide, test SDLS 02/89; Analysis type: modal analysis; Element type: planar.
2.9.2 Overview The 10 mm thick plate is fixed on one side and is loaded with its self weight only.
Thin lozenge-shaped plate fixed on one side Scale =1/10 01-0009SDLSB_FEM
Units I. S.
Geometry
Thickness: t = 0.01 m, Side: a = 1 m, α = 30° Points coordinates:
o A ( 0 ; 0 ; 0 ) o B ( a ; 0 ; 0 )
o C ( 0.5a ; 3 2 a ; 0 )
o D ( 1.5a ; 3 2 a ; 0 )
Materials properties Longitudinal elastic modulus: E = 2.1 x 1011 Pa, Poisson's ratio: ν = 0.3, Density: ρ = 7800 kg/m3.
Test 01-0006SDLLB_FEM
34
Boundary conditions
Outer: AB side fixed,
Inner: None.
Loading
External: None,
Internal: None.
2.9.3 Eigen mode frequencies relative to the α angle
Reference solution M. V. Barton formula for a lozenge of side "a" leads to the frequencies:
fj = ⋅⋅π 2a21
λi2
)1(12Et
2
2
ν−ρ where i = 1,2, or λi
2 = g(α).
α = 30° λ1
2 3.961 λ2
2 10.19
M. V. Barton noted the sensitivity of the result relative to the mode and the α angle. He acknowledged that the λi values were determined with a limited development of an insufficient order, which led to consider a reference value that is based on an experimental result, verified by an average of seven software that use the finite elements calculation method.
Finite elements modeling
Planar element: plate, imposed mesh,
961 nodes,
900 surface quadrangles.
Eigen mode shapes
2.9.4 Results sheet
Results comparison: eigen modes frequencies Solver Eigen mode type Units Reference AD 2010 Deviation CM2 Mode 1 Hz 9.8987 9.82 -0.80% CM2 Mode 2 Hz 25.4651 23.44 -7.95%
Test 01-0010SDLSB_FEM
35
2.10 Test No. 01-0010SDLSB_FEM: Thin lozenge-shaped plate fixed on one side (α = 45 °)
2.10.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLS 02/89;
Analysis type: modal analysis;
Element type: planar.
2.10.2 Overview The 10 mm thick plate is fixed on one side and is loaded with its self weight only.
Thin lozenge-shaped plate fixed on one side Scale =1/10 01-0010SDLSB_FEM
Units
I. S.
Geometry
Thickness: t = 0.01 m,
Side: a = 1 m,
α = 45°
Points coordinates: o A ( 0 ; 0 ; 0 ) o B ( a ; 0 ; 0 )
o C ( 22
a ; 22
a ; 0 )
o D (2
22 +a ;
22
a ; 0 )
Test 01-0006SDLLB_FEM
36
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: AB side fixed,
Inner: None.
Loading
External: None,
Internal: None.
2.10.3 Eigen mode frequencies relative to the α angle
Reference solution M. V. Barton formula for a lozenge of side "a" leads to the frequencies:
fj = ⋅⋅π 2a21
λi2
)1(12Et
2
2
ν−ρ where i = 1,2, or λi
2 = g(α).
α = 45° λ1
2 4.4502 λ2
2 10.56
M. V. Barton noted the sensitivity of the result relative to the mode and the α angle. He acknowledged that the λi values were determined with a limited development of an insufficient order, which led to consider a reference value that is based on an experimental result, verified by an average of seven software that use the finite elements calculation method.
Finite elements modeling
Planar element: plate, imposed mesh,
961 nodes,
900 surface quadrangles.
Eigen mode shapes
2.10.4 Results sheet
Results comparison: eigen mode frequencies Solver Eigen mode type Units Reference AD 2010 Deviation CM2 Mode 1 Hz 11.1212 11.28 1.43% CM2 Mode 2 Hz 26.3897 28.08 6.41%
Test 01-0011SDLLB_FEM
37
2.11 Test No. 01-0011SDLLB_FEM: Vibration mode of a thin piping elbow in plane (case 1)
2.11.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 14/89;
Analysis type: modal analysis (plane problem);
Element type: linear.
2.11.2 Overview A thin piping elbow with a radius of 1m has fixed ends and is loaded with its self weight only.
Vibration mode of a thin piping elbow in plane Scale = 1/7 Case 1 01-0011SDLLB_FEM
Units
I. S.
Geometry
Average radius of curvature: OA = R = 1 m,
Straight circular hollow section:
Outer diameter: de = 0.020 m,
Inner diameter: di = 0.016 m,
Section: A = 1.131 x 10-4 m2,
Flexure moment of inertia relative to the y-axis: Iy = 4.637 x 10-9 m4,
Flexure moment of inertia relative to z-axis: Iz = 4.637 x 10-9 m4,
Polar inertia: Ip = 9.274 x 10-9 m4.
Points coordinates (in m): o O ( 0 ; 0 ; 0 ) o A ( 0 ; R ; 0 ) o B ( R ; 0 ; 0 )
Test 01-0006SDLLB_FEM
38
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: Fixed at points A and B ,
Inner: None.
Loading
External: None,
Internal: None.
2.11.3 Eigen mode frequencies
Reference solution
The Rayleigh method applied to a thin curved beam is used to determine parameters such as:
in plane bending:
fj = 2
2i
R2 ⋅π
λ
AEIz
ρ where i = 1,2,
Finite elements modeling
Linear element: beam,
11 nodes,
10 linear elements.
Eigen mode shapes
2.11.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation CM2 In plane 1 Hz 119 120.09 0.92% CM2 In plane 2 Hz 227 227.10 0.04%
Test 01-0012SDLLB_FEM
39
2.12 Test No. 01-0012SDLLB_FEM: Vibration mode of a thin piping elbow in plane (case 2)
2.12.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 14/89; Analysis type: modal analysis (plane problem); Element type: linear.
2.12.2 Overview A thin piping elbow with a radius of 1m is extended by two straight elements of length L is loaded with its self weight only.
Vibration mode of a thin piping elbow Scale = 1/11 Case 2 01-0012SDLLB_FEM
Units I. S.
Geometry
Average radius of curvature: OA = R = 1 m, L = 0.6 m, Straight circular hollow section: Outer diameter de = 0.020 m, Inner diameter di = 0.016 m, Section: A = 1.131 x 10-4 m2, Flexure moment of inertia relative to the y-axis: Iy = 4.637 x 10-9 m4, Flexure moment of inertia relative to z-axis: Iz = 4.637 x 10-9 m4, Polar inertia: Ip = 9.274 x 10-9 m4. Points coordinates (in m):
o O ( 0 ; 0 ; 0 ) o A ( 0 ; R ; 0 ) o B ( R ; 0 ; 0 ) o C ( -L ; R ; 0 ) o D ( R ; -L ; 0 )
Test 01-0006SDLLB_FEM
40
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: o Fixed at points C and D o At A: translation restraint along y and z, o At B: translation restraint along x and z,
Inner: None.
Loading
External: None,
Internal: None.
2.12.3 Eigen mode frequencies Reference solution The Rayleigh method applied to a thin curved beam is used to determine parameters such as:
in plane bending:
fj = 2
2i
R2 ⋅π
λ
AEIz
ρ where i = 1,2,
Finite elements modeling
Linear element: beam,
23 nodes,
22 linear elements.
Eigen mode shapes
2.12.4 Results sheet Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation CM2 In plane 1 Hz 94 94.62 0.66% CM2 In plane 2 Hz 180 184.68 2.60%
Test 01-0013SDLLB_FEM
41
2.13 Test No. 01-0013SDLLB_FEM: Vibration mode of a thin piping elbow in plane (case 3)
2.13.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 14/89;
Analysis type: modal analysis (plane problem);
Element type: linear.
2.13.2 Overview A thin piping elbow with a radius of 1m is extended by two straight elements of length L is loaded with its self weight only.
Vibration mode of a thin piping elbow Scale = 1/12 Case 3 01-0013SDLLB_FEM
Units I. S.
Geometry
Average radius of curvature: OA = R = 1 m,
Straight circular hollow section:
Outer diameter: de = 0.020 m,
Inner diameter: di = 0.016 m,
Section: A = 1.131 x 10-4 m2,
Flexure moment of inertia relative to the y-axis: Iy = 4.637 x 10-9 m4,
Flexure moment of inertia relative to z-axis: Iz = 4.637 x 10-9 m4,
Polar inertia: Ip = 9.274 x 10-9 m4.
Points coordinates (in m): o O ( 0 ; 0 ; 0 ) o A ( 0 ; R ; 0 ) o B ( R ; 0 ; 0 ) o C ( -L ; R ; 0 ) o D ( R ; -L ; 0 )
Test 01-0006SDLLB_FEM
42
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: o Fixed at points C and Ds, o At A: translation restraint along y and z, o At B: translation restraint along x and z,
Inner: None.
Loading
External: None,
Internal: None.
2.13.3 Eigen mode frequencies
Reference solution The Rayleigh method applied to a thin curved beam is used to determine parameters such as:
in plane bending:
fj = 2
2i
R2 ⋅π
λ
AEIz
ρ where i = 1,2,
Finite elements modeling
Linear element: beam,
41 nodes,
40 linear elements.
Eigen mode shapes
2.13.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation CM2 In plane 1 Hz 25.300 24.961 -1.34% CM2 In plane 2 Hz 27.000 26.710 -1.07%
Test 01-0014SDLLB_FEM
43
2.14 Test No. 01-0014SDLLB_FEM: Thin circular ring hang from an elastic element
2.14.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 13/89;
Analysis type: modal analysis, plane problem;
Element type: linear.
2.14.2 Overview Search for the first eigen modes frequencies of a circular ring hang from an elastic element and loaded with its own weight only.
Thin circular ring hang from an elastic element Scale = 1/1 01-0014SDLLB_FEM
Units
I. S.
Geometry
Average radius of curvature: OB = R = 0.1 m,
Length of elastic element: AB = 0.0275 m ;
Test 01-0014SDLLB_FEM
44
Straight rectangular section: o Ring Thickness: h = 0.005 m, Width: b = 0.010 m, Section: A = 5 x 10-5 m2, Flexure moment of relative to the vertical axis: I = 1.042 x 10-10 m4,
o Elastic element Thickness: h = 0.003 m, Width: b = 0.010 m, Section: A = 3 x 10-5 m2, Flexure moment of inertia relative to the vertical axis: I = 2.25 x 10-11 m4,
Points coordinates:
o O ( 0 ; 0 ), o A ( 0 ; -0.0725 ), o B ( 0 ; -0.1 ).
Materials properties
Longitudinal elastic modulus: E = 7.2 x 1010 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 2700 kg/m3.
Boundary conditions
Outer: Fixed in A,
Inner: None.
Loading
External: None,
Internal: None.
2.14.3 Eigen mode frequencies
Reference solutions
The reference solution was established from experimental results of a mass manufactured aluminum ring.
Finite elements modeling
Linear element: beam,
43 nodes,
43 linear elements.
Test 01-0014SDLLB_FEM
45
Eigen mode shapes
2.14.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Eigen mode order Units Reference AD 2010 Deviati
on CM2 Mode 1 Asymmetrical Hz 28.80 28.81 0.03% CM2 Mode 2 Symmetrical Hz 189.30 189.69 0.21% CM2 Mode 3 Asymmetrical Hz 268.80 269.38 0.22% CM2 Mode 4 Asymmetrical Hz 641.00 642.15 0.18% CM2 Mode 5 Symmetrical Hz 682.00 683.9 0.28% CM2 Mode 6 Asymmetrical Hz 1063.00 1065.73 0.26%
Test 01-0015SSLLB_FEM
46
2.15 Test No. 01-0015SSLLB_FEM: Double fixed beam with a spring at mid span
2.15.1 Description sheet
Reference: internal GRAITEC test;
Analysis type: linear static;
Element type: linear.
2.15.2 Overview Consider the double fixed beam described below. This beam consists of four elements of the length l having identical characteristics.
Units
I. S.
Geometry
l = 1 m
S=0.01 m2
I = 0.0001 m4
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer:
Fixed at ends x = 0 and x = 4 m,
Elastic support with k = EI/l rigidity
Inner: None.
Loading
External: Punctual load P = -10000 N at x = 2m,
Internal: None.
Test 01-0015SSLLB_FEM
47
2.15.3 Displacement of the model in the linear elastic range
Reference solution
The reference vertical displacement v3, is calculated at the middle of the beam at x = 2 m.
Rigidity matrix of a plane beam:
[ ]
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−−
−
−
−
=
llll
llll
ll
lll
llll
ll
EIEIEIEI
EIEIEIEI
EIEIEIEI
EIEIEIEI
460260
61206120
00l
ES00ES
260460
61206120
00ES-00ES
K
22
2323
22
2323
e
Given the symmetry / X and load of the structure, it is unnecessary to consider the degrees of freedom associated with normal work (u2, u3, u4).
The same symmetry allows the deduction of:
v2 = v4
β2 = -β4
β3 = 0
( )( )( )( )( )( )654321
000
00
4626
612612
268026
612024612
268026
6120124612
268026
612024612
2646
612612
5
5
1
1
5
5
4
4
3
3
2
2
1
1
22
22
22
22
22
22
22
22
22
22
⎪⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
−=
⎪⎪⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−−
−
−−−
−
−⎟⎠
⎞⎜⎝
⎛ +−−
−
−−−
−
−
MR
P
MR
v
v
v
v
v
EI
β
β
β
β
β
llll
llll
lllll
lllll
lllll
llllll
lllll
lllll
llll
llll
33
333
333
333
33
Test 01-0015SSLLB_FEM
48
The elementary rigidity matrix of the spring in its local axis system, [ ])()(
1111
6
35 U
UEIk ⎥⎦
⎤⎢⎣
⎡−
−=
l, must be expressed in the
global axis system by means of the rotation matrix (90° rotation):
[ ]
( )( )( )( )( )( )6
6
6
3
3
3
5
000000010010000000000000010010000000
β
β
vu
vu
EIK
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
=l
→ 344332 43 0826 vvllll
−=⇒=++ βββ
→ 344332332 024612 vvvv =⇒=+−−
lllβ
→ y)unnecessar(usually 026826244423222 vvvv =⇒=+−++ βββ
lllll
(3) → ( ) m 10 11905.03
612124612 032
3
34243332223−−=
+−=⇒−=−−⎟
⎠
⎞⎜⎝
⎛ ++−−EIl
PvEIPvvv
llllllββ
Finite elements modeling
Linear element: beam, imposed mesh,
6 nodes,
4 linear elements + 1 spring,
Deformed shape
Double fixed beam with a spring at mid span Deformed
Note: the displacement is expressed here in μm
2.15.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 Middle of the beam Mm -0.11905 -0.11905 0.00%
Test 01-0016SDLLB_FEM
49
2.16 Test No. 01-0016SDLLB_FEM: Double fixed beam
2.16.1 Description sheet
Reference: internal GRAITEC test (beams theory);
Analysis type: static linear, modal analysis;
Element type: linear.
2.16.2 Overview Consider the double fixed beam described below. This beam consists of eight elements of the length l having identical characteristics.
Units
I. S.
Geometry
Length: l = 16 m,
Axial section: S=0.06 m2
Inertia I = 0.0001 m4
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 N/m2,
Poisson's ratio: ν = 0.3,
Density: ρ = 7850 kg/m3
Boundary conditions
Outer: Fixed at both ends x = 0 and x = 8 m,
Inner: None.
Loading
External: Punctual load P = -50000 N at x = 4m,
Internal: None.
Test 01-0016SDLLB_FEM
50
2.16.3 Displacement of the model in the linear elastic range
Reference solution
The reference vertical displacement v5, is calculated at the middle of the beam at x = 2 m.
m 05079.00001.0111.2192
1650000192
33
5 =××
×==
EEIPlv
Finite elements modeling
Linear element: beam, imposed mesh,
9 nodes,
8 elements.
Deformed shape
Double fixed beam Deformed
2.16.4 Eigen mode frequencies of the model in the linear elastic range
Reference solution
Knowing that the first four eigen mode frequencies of a double fixed beam are given by the following formula:
SIE
Lf nn .
...2 2
2
ρπχ
= where for the first 4 eigen modes frequencies
⎪⎪
⎩
⎪⎪
⎨
⎧
→=
→=
→=
→=
Hz 26.228=f 8.199
Hz 15.871=f 9.120
Hz 8.095=f 67.61
Hz 2.937=f 37.22
424
323
222
121
χ
χ
χ
χ
Finite elements modeling
Linear element: beam, imposed mesh,
9 nodes,
8 elements.
Test 01-0016SDLLB_FEM
51
Modal deformations
Double fixed beam Mode 1
Double fixed beam Mode 2
Double fixed beam Mode 3
Test 01-0016SDLLB_FEM
52
Double fixed beam
Mode 4
2.16.5 Results sheet
1 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation
CM2 Middle of the beam m -0.05079 -0.05079 0.00%
2 Results comparison: eigen mode frequencies
Solver Eigen modes Units Reference AD 2010 Deviation
CM2 1 Hz 2.937 2.937 0.00%
CM2 2 Hz 8.095 8.087 -0.10%
CM2 3 Hz 15.870 15.789 -0.51%
CM2 4 Hz 26.228 25.758 -1.79%
Test 01-0017SDLLB_FEM
53
2.17 Test No. 01-0017SDLLB_FEM: Short beam on simple supports (on the neutral axis)
2.17.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 01/89;
Analysis type: modal analysis (plane problem);
Element type: linear.
2.17.2 Overview Search for the first eigen mode frequencies of a beam on simple supports (the supports are located on the neutral axis).
Short beam on simple supports on the neutral axis Scale = 1/6 01-0017SDLLB_FEM
Units
I. S.
Geometry
Height: h = 0.2 m,
Length: l = 1 m,
Width: b = 0.1 m,
Section: A = 2 x 10-2 m4,
Flexure moment of inertia relative to z-axis: Iz = 6.667 x 10-5 m4.
Test 01-0017SDLLB_FEM
54
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: o Hinged at A (null horizontal and vertical displacements), o Simple support in B.
Inner: None.
Loading
External: None.
Internal: None.
2.17.3 Eigen modes frequencies
Reference solution
The bending beams equation gives, when superimposing, the effects of simple bending, shear force deformations and rotation inertia, Timoshenko formula.
The reference eigen modes frequencies are determined by a numerical simulation of this equation, independent of any software.
The eigen frequencies in tension-compression are given by:
fi = ⋅π
λl2
i ρE
where λi = 2
)1i2( −
Finite elements modeling
Linear element: S beam, imposed mesh,
10 nodes,
9 linear elements.
Eigen mode shapes
Test 01-0017SDLLB_FEM
55
2.17.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation
CM2 Mode 1 Hz 431.555 437.118 1.29% CM2 Mode 2 Hz 1265.924 1264.319 -0.13% CM2 Mode 3 Hz 1498.295 1537.156 2.59% CM2 Mode 4 Hz 2870.661 2911.456 1.42% CM2 Mode 5 Hz 3797.773 3754.542 -1.14% CM2 Mode 6 Hz 4377.837 4281.235 -2.21%
Test 01-0018SDLLB_FEM
56
2.18 Test No. 01-0018SDLLB_FEM: Short beam on simple supports (eccentric)
2.18.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 01/89;
Analysis type: modal analysis, (plane problem);
Element type: linear.
2.18.2 Overview Search for the first eigen mode frequencies of a beam on simple supports (the supports are eccentric relative to the neutral axis).
Short beam on simple supports (eccentric) Scale = 1/5 01-0018SDLLB_FEM
Units
I. S.
Geometry
Height: h = 0.2m,
Length: l = 1 m,
Width: b = 0.1 m,
Section: A = 2 x 10-2 m4,
Flexure moment of inertia relative to z-axis: Iz = 6.667 x 10-5 m4.
Test 01-0018SDLLB_FEM
57
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: o Hinged at A (null horizontal and vertical displacements), o Simple support at B.
Inner: None.
Loading
External: None.
Internal: None.
2.18.3 Eigen modes frequencies
Reference solution
The problem has no analytical solution, the solution is determined by averaging several software: Timoshenko model with shear force deformation effects and rotation inertia. The bending modes and the traction-compression are coupled.
Finite elements modeling
Linear element: S beam, imposed mesh,
10 nodes,
9 linear elements.
Eigen modes shape
Short beam on simple supports (eccentric) Mode 1
Test 01-0018SDLLB_FEM
58
Short beam on simple supports (eccentric)
Mode 2
Short beam on simple supports (eccentric) Mode 3
Short beam on simple supports (eccentric) Mode 4
Test 01-0018SDLLB_FEM
59
Short beam on simple supports (eccentric)
Mode 5
2.18.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference Reference
margin AD 2010 Deviati
on CM2 Mode 1 Hz 392.8 ± 2.5% 393.70 0.23% CM2 Mode 2 Hz 902.2 ± 5% 945.35 4.78% CM2 Mode 3 Hz 1591.9 ± 3% 1595.94 0.25% CM2 Mode 4 Hz 2629.2 ± 5% 2526.22 -3.92%CM2 Mode 5 Hz 3126.2 ± 4% 3118.91 -0.23%
Test 01-0019SDLSB_FEM
60
2.19 Test No. 01-0019SDLSB_FEM: Thin square plate fixed on one side
2.19.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLS 01/89;
Analysis type: modal analysis;
Element type: planar.
2.19.2 Overview Research of the first eigen modes frequencies of a thin square plate fixed on one side.
Thin square plate fixed on one side Scale = 1/6 01-0019SDLSB_FEM
Units I. S.
Geometry
Side: a = 1 m,
Thickness: t = 1 m,
Points coordinates in m: o A (0 ;0 ;0) o B (1 ;0 ;0) o C (1 ;1 ;0) o D (0 ;1 ;0)
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Test 01-0019SDLSB_FEM
61
Boundary conditions
Outer: Edge AD fixed.
Inner: None.
Loading
External: None.
Internal: None.
2.19.3 Eigen modes frequencies
Reference solution
M. V. Barton formula for a square plate with side "a", leads to:
fj = 2a2
1⋅π
λi2
)1(12Et
2
2
ν−ρ where i = 1,2, . . .
i 1 2 3 4 5 6 λi 3.492 8.525 21.43 27.33 31.11 54.44
Finite elements modeling
Planar element: shell,
959 nodes,
900 planar elements.
Eigen mode shapes
Thin square plate fixed on one side Mode 1
Test 01-0019SDLSB_FEM
62
Thin square plate fixed on one side
Mode 2
Thin square plate fixed on one side Mode 3
2.19.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation
CM2 Mode 1 Hz 8.7266 8.671 -0.64% CM2 Mode 2 Hz 21.3042 21.218 -0.40% CM2 Mode 3 Hz 53.5542 53.130 -0.79% CM2 Mode 4 Hz 68.2984 67.737 -0.82% CM2 Mode 5 Hz 77.7448 77.147 -0.77% CM2 Mode 6 Hz 136.0471 134.655 -1.02%
Test 01-0020SDLSB_FEM
63
2.20 Test No. 01-0020SDLSB_FEM: Rectangular thin plate simply supported on its perimeter
2.20.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLS 03/89;
Analysis type: modal analysis;
Element type: planar.
2.20.2 Overview Search for the first eigen mode frequencies of a thin rectangular plate fixed simply supported on its perimeter.
Rectangular thin plate simply supported on its perimeter Scale = 1/8 01-0020SDLSB_FEM
Units
I. S.
Geometry
Length: a = 1.5 m,
Width: b = 1 m,
Thickness: t = 0.01 m,
Test 01-0020SDLSB_FEM
64
Points coordinates in m: o A (0 ;0 ;0) o B (0 ;1.5 ;0) o C (1 ;1.5 ;0) o D (1 ;0 ;0)
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: o Simple support on all sides, o For the modeling: hinged at A, B and D.
Inner: None.
Loading
External: None.
Internal: None.
2.20.3 Eigen modes frequencies
Reference solution
M. V. Barton formula for a rectangular plate with supports on all four sides, leads to:
fij = 2π
[ ( ai
)2 + ( bj
)2] )1(12
Et2
2
ν−ρ
where: i = number of half-length of wave along y ( dimension a)
j = number of half-length of wave along x ( dimension b)
Finite elements modeling
Planar element: shell,
496 nodes,
450 planar elements.
Test 01-0020SDLSB_FEM
65
Eigen mode shapes
2.20.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type i j
Units Reference AD 2010 Deviation
CM2 1 1 Hz 35.63 35.580 -0.14% CM2 2 1 Hz 68.51 68.292 -0.32% CM2 1 2 Hz 109.62 109.982 0.33% CM2 3 1 Hz 123.32 123.021 -0.24% CM2 2 2 Hz 142.51 141.976 -0.37% CM2 3 2 Hz 197.32 195.545 -0.90%
Test 01-0021SFLLB_FEM
66
2.21 Test No. 01-0021SFLLB_FEM: Cantilever beam in Eulerian buckling
2.21.1 Description sheet
Reference: internal GRAITEC test (Euler theory);
Analysis type: Eulerian buckling;
Element type: linear.
2.21.2 Overview
Units
I. S.
Geometry
L = 10 m
S=0.01 m2
I = 0.0002 m4
Materials properties
Longitudinal elastic modulus: E = 2.0 x 1010 N/m2,
Poisson's ratio: ν = 0.1.
Boundary conditions
Outer: Fixed at end x = 0,
Inner: None.
Loading
External: Punctual load P = -100000 N at x = L,
Internal: None.
Test 01-0021SFLLB_FEM
67
2.21.3 Critical load on node 5
Reference solution
The reference critical load established by Euler is:
98696.010000098696N 98696
L4EIP 2
2
critique ==λ⇒=π
=
Finite elements modeling
Planar element: beam, imposed mesh,
5 nodes,
4 elements.
Deformed shape
2.21.4 Results sheet
Results comparison: critical load
Solver Positioning Units Reference AD 2010 Deviation CM2 On node 5 N -98696 -98699.278 0.00%
Test 01-0022SDLSB_FEM
68
2.22 Test No. 01-0022SDLSB_FEM: Annular thin plate fixed on a hub (repetitive circular structure)
2.22.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLS 04/89;
Analysis type: modal analysis;
Element type: planar element.
2.22.2 Overview Search for the eigen mode frequencies of a thin annular plate fixed on a hub.
Annular thin plate fixed on a hub (repetitive circular structure) Scale = 1/3 01-0022SDLSB_FEM
Units
I. S.
Geometry
Inner radius: Ri = 0.1 m,
Outer radius: Re = 0.2 m,
Thickness: t = 0.001 m.
Material properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Test 01-0022SDLSB_FEM
69
Boundary conditions
Outer: Fixed on a hub at any point r = Ri.
Inner: None.
Loading
External: None.
Internal: None.
2.22.3 Eigen modes frequencies
Reference solution
The solution of determining the frequency based on Bessel functions leads to the following formula:
fij = 1
2πRe2 λij
2 Et2
12ρ(1-ν2)
where: i = the number of nodal diameters
j = the number of nodal circles
and λij2 such as:
j \ i 0 1 2 3 0 13.0 13.3 14.7 18.5 1 85.1 86.7 91.7 100
Finite elements modeling
Planar element: plate,
360 nodes,
288 planar elements.
2.22.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type i j
Corresponding eigen mode in AD 2010
Units Reference AD 2010 Deviation
CM2 0 0 1 Hz 79.26 79.05 -0.26% CM2 0 1 18 Hz 518.85 521.84 0.58% CM2 1 0 2 Hz 81.09 80.52 -0.70% CM2 1 1 20 Hz 528.61 529.49 0.17% CM2 2 0 4 Hz 89.63 88.43 -1.34% CM2 2 1 22 Hz 559.09 552.43 -1.19% CM2 3 0 7 Hz 112.79 110.27 -2.23% CM2 3 1 25 Hz 609.70 593.83 -2.60%
Test 01-0023SDLLB_FEM
70
2.23 Test No. 01-0023SDLLB_FEM: Bending effects of a symmetrical portal frame
2.23.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 01/89;
Analysis type: modal analysis;
Element type: linear.
2.23.2 Overview Search for the first eigen mode frequencies of a symmetrical portal frame with fixed supports.
Bending effects of a symmetrical portal frame Scale = 1/5 01-0023SDLLB_FEM
Test 01-0023SDLLB_FEM
71
Units
I. S.
Geometry
Straight rectangular sections for beams and columns:
Thickness: h = 0.0048 m,
Width: b = 0.029 m,
Section: A = 1.392 x 10-4 m2,
Flexure moment of inertia relative to z-axis: Iz = 2.673 x 10-10 m4,
Points coordinates in m: A B C D E F x -0.30 0.30 -0.30 0.30 -0.30 0.30 y 0 0 0.36 0.36 0.81 0.81
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: Fixed at A and B,
Inner: None.
Loading
External: None.
Internal: None.
2.23.3 Eigen modes frequencies
Reference solution
Dynamic radius method (slender beams theory).
Finite elements modeling
Linear element: beam,
60 nodes,
60 linear elements.
Test 01-0023SDLLB_FEM
72
Deformed shape
Bending effects of a symmetrical portal frame Scale = 1/7 Mode 13
2.23.4 Results sheet
Results comparison: eigen mode frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation
CM2 1 antisymmetric Hz 8.8 8.8 0.00% CM2 2 antisymmetric Hz 29.4 29.4 0.00% CM2 3 symmetric Hz 43.8 43.8 0.00% CM2 4 symmetric Hz 56.3 56.3 0.00% CM2 5 antisymmetric Hz 96.2 96.1 -0.10% CM2 6 symmetric Hz 102.6 102.7 0.10% CM2 7 antisymmetric Hz 147.1 147.1 0.00% CM2 8 symmetric Hz 174.8 174.9 0.06% CM2 9 antisymmetric Hz 178.8 178.9 0.06% CM2 10 antisymmetric Hz 206 206.2 0.10% CM2 11 symmetric Hz 266.4 266.6 0.08% CM2 12 antisymmetric Hz 320 319.95 -0.02% CM2 13 symmetric Hz 335 334.96 -0.01%
Test 01-0024SSLLB_FEM
73
2.24 Test No. 01-0024SSLLB_FEM: Slender beam on two fixed supports
2.24.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 01/89;
Analysis type: linear static;
Element type: linear.
2.24.2 Overview A straight slender beam with fixed ends is loaded with a uniform load, several punctual loads and a torque.
Slender beam on two fixed supports Scale = 1/4 01-0024SSLLB_FEM
Test 01-0024SSLLB_FEM
74
Units
I. S.
Geometry
Length: L = 1 m,
Beam inertia: I = 1.7 x 10-8 m4.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa.
Boundary conditions
Outer: Fixed at A and B,
Inner: None.
Loading
External: o Uniformly distributed load from A to B: py = p = -24000 N/m, o Punctual load at D: Fx = F1 = 30000 N, o Torque at D: Cz = C = -3000 Nm, o Punctual load at E: Fx = F2 = 10000 N, o Punctual load at E: Fy = F = -20000 N.
Internal: None.
Test 01-0024SSLLB_FEM
75
2.24.3 Shear force at G
Reference solution
Analytical solution:
Shear force at G: VG
VG = 0.216F – 1.26 LC
Finite elements modeling
Linear element: beam,
5 nodes,
4 linear elements.
Results shape
Slender beam on two fixed supports Scale = 1/5
Shear force
Test 01-0024SSLLB_FEM
76
2.24.4 Bending moment in G
Reference solution
Analytical solution:
Bending moment at G: MG
MG = pL2
24 - 0.045LF – 0.3C
Finite elements modeling
Linear element: beam,
5 nodes,
4 linear elements.
Results shape
Slender beam on two fixed supports Scale = 1/5
Bending moment
Test 01-0024SSLLB_FEM
77
2.24.5 Vertical displacement at G
Reference solution
Analytical solution:
Vertical displacement at G: vG
vG = pl4
384EI + 0.003375FL3
EI + 0.015CL2
EI
Finite elements modeling
Linear element: beam,
5 nodes,
4 linear elements.
Results shape
Slender beam on two fixed supports Scale = 1/4
Deformed
Test 01-0024SSLLB_FEM
78
2.24.6 Horizontal reaction at A
Reference solution
Analytical solution:
Horizontal reaction at A: HA HA = -0.7F1 –0.3F2
Finite elements modeling
Linear element: beam,
5 nodes,
4 linear elements.
2.24.7 Results sheet
1 Results comparison: shear force
Solver Positioning Units Reference AD 2010 Deviation CM2 In G N -540 -540 0.00%
2 Results comparison: bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 In G Nm -2800 -2800 0.00%
3 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In G cm -4.90 -4.90 0.00%
4 Results comparison: horizontal reaction
Solver Positioning Units Reference AD 2010 Deviation CM2 In A N 24000 24000 0.00%
Test 01-0025SSLLB_FEM
79
2.25 Test No. 01-0025SSLLB_FEM: Slender beam on three supports
2.25.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 03/89;
Analysis type: static (plane problem);
Element type: linear.
2.25.2 Overview A straight slender beam on three supports is loaded with two punctual loads.
Slender beam on three supports Scale = 1/49 01-0025SSLLB_FEM
Units I. S.
Geometry Length: L = 3 m,
Beam inertia: I = 6.3 x 10-4 m4.
Materials properties Longitudinal elastic modulus: E = 2.1 x 1011 Pa.
Boundary conditions Outer:
o Hinged at A, o Elastic support at B (Ky = 2.1 x 106 N/m), o Simple support at C.
Inner: None.
Test 01-0025SSLLB_FEM
80
Loading
External: 2 punctual loads F = Fy = -42000N.
Internal: None.
2.25.3 Bending moment at B Reference solution The resolution of the hyperstatic system of the slender beam leads to:
k = Ky3LEI6
Bending moment at B: MB
MB = ± 2L
)k8(F)k26(
++−
Finite elements modeling
Linear element: beam,
5 nodes,
4 linear elements.
Results shape
Slender beam on three supports Scale = 1/49
Bending moment
2.25.4 Reaction in B Reference solution
Compression force in the spring: VB
VB = -11F8 + k
Finite elements modeling
Linear element: beam,
5 nodes,
4 linear elements.
Test 01-0025SSLLB_FEM
81
2.25.5 Vertical displacement at B
Reference solution
Deflection at the spring location: vB
vB = 11F
Ky(8 + k)
Finite elements modeling
Linear element: beam,
5 nodes,
4 linear elements.
Results shape
Slender beam on three supports Deformed
2.25.6 Results sheet
1 Results comparison: bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 In B Nm -63000 -63000 0.00%
2 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In B cm -1.00 -1.00 0.00%
3 Results comparison: reaction
Solver Positioning Units Reference AD 2010 Deviation CM2 In B N -21000 -21000 0.00%
Test 01-0026SSLLB_FEM
82
2.26 Test No. 01-0026SSLLB_FEM: Bimetallic: Fixed beams connected to a stiff element
2.26.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 05/89;
Analysis type: linear static;
Element type: linear.
2.26.2 Overview Two beams fixed at one end and rigidly connected to an undeformable beam is loaded with a punctual load.
Fixed beams connected to a stiff element Scale = 1/10 01-0026SSLLB_FEM
Units
I. S.
Geometry
Lengths: o L = 2 m, o l = 0.2 m,
Beams inertia moment: I = (4/3) x 10-8 m4,
The beam sections are squared, of side: 2 x 10-2 m.
Test 01-0026SSLLB_FEM
83
Materials properties
Longitudinal elastic modulus: E = 2 x 1011 Pa.
Boundary conditions
Outer: Fixed in A and C,
Inner: The tangents to the deflection of beams AB and CD at B and D remain horizontal; practically, we restraint translations along x and z at nodes B and D.
Loading
External: In D: punctual load F = Fy = -1000N.
Internal: None.
2.26.3 Deflection at B and D
Reference solution
The theory of slender beams bending (Euler-Bernouilli formula) leads to a deflection at B and D:
The resolution of the hyperstatic system of the slender beam leads to:
vB = vD = FL3
24EI
Finite elements modeling
Linear element: beam,
4 nodes,
3 linear elements.
Results shape
Fixed beams connected to a stiff element Scale = 1/10 Deformed
Test 01-0026SSLLB_FEM
84
2.26.4 Vertical reaction at A and C
Reference solution
Analytical solution.
Finite elements modeling
Linear element: beam,
4 nodes,
3 linear elements.
2.26.5 Bending moment at A and C
Reference solution
Analytical solution.
Finite elements modeling
Linear element: beam,
4 nodes,
3 linear elements
2.26.6 Results sheet
1 Results comparison: deflection
Solver Positioning Units Reference AD 2010 Deviation CM2 In B m 0.125 0.125 0.00% CM2 In D m 0.125 0.125 0.00%
2 Results comparison: vertical reaction
Solver Positioning Units Reference AD 2010 Deviation CM2 In A N -500 -500 0.00% CM2 In C N -500 -500 0.00%
3 Results comparison: Bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 In A Nm 500 500.01 0.00% CM2 In C Nm 500 500.01 0.00%
Test 01-0027SSLLB_FEM
85
2.27 Test No. 01-0027SSLLB_FEM: Fixed thin arc in planar bending
2.27.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 06/89;
Analysis type: static linear (plane problem);
Element type: linear.
2.27.2 Overview Arc of a circle fixed at one end, subjected to two punctual loads and a torque at its free end.
Fixed thin arc in planar bending Scale = 1/24
01-0027SSLLB_FEM
Test 01-0027SSLLB_FEM
86
Units
I. S.
Geometry
Medium radius: R = 3 m ,
Circular hollow section: o de = 0.02 m, o di = 0.016 m, o A = 1.131 x 10-4 m2, o Ix = 4.637 x 10-9 m4.
Materials properties
Longitudinal elastic modulus: E = 2 x 1011 Pa.
Boundary conditions
Outer: Fixed in A.
Inner: None.
Loading
External: At B:
o punctual load F1 = Fx = 10 N, o punctual load F2 = Fy = 5 N, o bending moment about Oz, Mz = 8 Nm.
Internal: None.
2.27.3 Displacements at B
Reference solution
At point B:
displacement parallel to Ox: u = R2
4EI [F1πR + 2F2R + 4Mz]
displacement parallel to Oy: v = R2
4EI [2F1πR + (3π - 8)F2R + 2(π - 2)Mz]
rotation around Oz: θ = R
4EI [4F1R + 2(π - 2)F2R + 2πMz]
Finite elements modeling
Linear element: beam,
31 nodes,
30 linear elements.
Test 01-0027SSLLB_FEM
87
Results shape
Fixed thin arc in planar bending Scale = 1/19 Deformed
2.27.4 Results sheet
1 Results comparison: horizontal displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In B m 0.3791 0.37891 -0.05%
2 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In B m 0.2417 0.2417 0.00%
3 Results comparison: rotation about Z-axis
Solver Positioning Units Reference AD 2010 Deviation CM2 In B rad -0.1654 -0.1654 0.00%
Test 01-0028SSLLB_FEM
88
2.28 Test No. 01-0028SSLLB_FEM: Fixed thin arc in out of plane bending
2.28.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 07/89;
Analysis type: static linear;
Element type: linear.
2.28.2 Overview Arc of a circle fixed at one end, loaded with a punctual force at its free end perpendicular to the plane.
Fixed thin arc in out of plane bending Scale = 1/6 01-0028SSLLB_FEM
Units I. S.
Geometry
Medium radius: R = 1 m ,
Circular hollow section: o de = 0.02 m, o di = 0.016 m, o A = 1.131 x 10-4 m2, o Ix = 4.637 x 10-9 m4.
Materials properties
Longitudinal elastic modulus: E = 2 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Test 01-0028SSLLB_FEM
89
Boundary conditions
Outer: Fixed at A.
Inner: None.
Loading
External: Punctual force in B perpendicular on the plane: Fz = F = 100 N.
Internal: None.
2.28.3 Displacements at B
Reference solution Displacement out of plane at point B:
uB = FR3
EIx [ π4 +
EIx KT
(3π4 - 2)]
where KT is the torsional rigidity for a circular section (torsion constant is 2Ix).
KT = 2GIx = EIx
1 + ν ⇒ uB = FR3
EIx [ π4 + (1 + ν) (
3π4 - 2)]
Finite elements modeling
Linear element: beam,
46 nodes,
45 linear elements.
2.28.4 Moments at θ = 15°
Reference solution
Torsion moment: Mx’ = Mt = FR(1 - sinθ)
Bending moment: Mz’ = Mf = -FRcosθ
Finite elements modeling
Linear element: beam,
46 nodes,
45 linear elements.
2.28.5 Results sheet
1 Results comparison: displacement out of plane
Solver Positioning Units Reference AD 2010 Deviation CM2 In B m 0.13462 0.13516 0.40%
2 Results comparison: torsion moment
Solver Positioning Units Reference AD 2010 Deviation CM2 In θ = 15° Nm 74.1180 74.1220 0.01%
3 Results comparison: bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 In θ = 15° Nm -96.5925 -96.5779 -0.02%
Test 01-0029SSLLB_FEM
90
2.29 Test No. 01-0029SSLLB_FEM: Double hinged thin arc in planar bending
2.29.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 08/89;
Analysis type: static linear (plane problem);
Element type: linear.
2.29.2 Overview Double hinged thin arc in planar bending Scale = 1/8
01-0029SSLLB_FEM
Test 01-0029SSLLB_FEM
91
Units
I. S.
Geometry
Medium radius: R = 1 m ,
Circular hollow section: o de = 0.02 m, o di = 0.016 m, o A = 1.131 x 10-4 m2, o Ix = 4.637 x 10-9 m4.
Materials properties
Longitudinal elastic modulus: E = 2 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: o Hinge at A, o At B: allowed rotation along z, vertical displacement restrained along y.
Inner: None.
Loading
External: Punctual load at C: Fy = F = - 100 N.
Internal: None.
2.29.3 Displacements at A, B and C
Reference solution
Rotation about z-axis
θA = - θB = ( π2 - 1)
FR22EI
Displacement;
Vertical at C: vC = π8
FREA + (
3π4 - 2)
FR3
2EI
Horizontal at B: uB = FR
2EA - FR3
2EI
Finite elements modeling
Linear element: beam,
37 nodes,
36 linear elements.
Test 01-0029SSLLB_FEM
92
Displacements shape
Fixed thin arc in planar bending Scale = 1/11 Deformed
2.29.4 Results sheet
1 Results comparison: rotation about Z-axis
Solver Positioning Units Reference AD 2010 Deviation CM2 In A rad 0.030774 0.030778 0.01% CM2 In B rad -0.030774 -0.030778 0.01%
2 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In C cm -1.9206 -1.9202 -0.02%
3 Results comparison: horizontal displacement
Solver Model Units Reference AD 2010 Deviation CM2 In B cm 5.3912 5.386 -0.10%
Test 01-0030SSLLB_FEM
93
2.30 Test No. 01-0030SSLLB_FEM: Portal frame with lateral connections
2.30.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 10/89;
Analysis type: static linear;
Element type: linear.
2.30.2 Overview Portal frame with lateral connections Scale = 1/21
01-0030SSLLB_FEM
Units
I. S.
Geometry
Beam Length Moment of inertia AB lAB = 4 m IAB =
643 x 10-8 m4
AC lAC = 1 m IAC = 112 x 10-8 m4
AD lAD = 1 m IAD = 112 x 10-8 m4
AE lAE = 2 m IAE = 43 x 10-8 m4
Test 01-0030SSLLB_FEM
94
G is in the middle of DA.
The beams have square sections: o AAB = 16 x 10-4 m o AAD = 1 x 10-4 m o AAC = 1 x 10-4 m o AAE = 4 x 10-4 m
Materials properties
Longitudinal elastic modulus: E = 2 x 1011 Pa,
Boundary conditions
Outer: o Fixed at B, D and E, o Hinge at C,
Inner: None.
Loading
External: o Punctual force at G: Fy = F = - 105 N, o Distributed load on beam AD: p = - 103 N/m.
Internal: None.
2.30.3 Displacements at A
Reference solution
Rotation at A about z-axis:
We say: kAn = EIAnlAn
where n = B, C, D or E
K = kAB + kAD + kAE + 34 kAC
rAn = kAnK
C1 = FlAD
8 - plAB
2
12
θ = C14K
Finite elements modeling
Linear element: beam,
6 nodes,
5 linear elements.
Test 01-0030SSLLB_FEM
95
Displacements shape
Portal frame with lateral connections Deformed
2.30.4 Moments in A
Reference solution
MAB = plAB
2
12 + rAB x C1
MAD = - FlAD
8 + rAD x C1
MAE = rAE x C1
MAC = rAC x C1
Finite elements modeling
Linear element: beam,
6 nodes,
5 linear elements
2.30.5 Results sheet
Displacement results comparison: rotation θ about z-axis
Solver Positioning Units Reference AD 2010 Deviation CM2 In A rad -0.227118 -0.227401 0.12%
Moments results comparison: bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 In A (MAB) Nm 11023.72 11020.998 -0.02% CM2 In A (MAC) Nm 113.559 113.7044 0.13% CM2 In A (MAD) Nm 12348.588 12347.476 -0.01% CM2 In A (MAE) Nm 1211.2994 1212.7730 0.12%
Test 01-0031SSLLB_FEM
96
2.31 Test No. 01-0031SSLLB_FEM: Truss with hinged bars under a punctual load
2.31.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 11/89;
Analysis type: static linear (plane problem);
Element type: linear.
2.31.2 Overview Truss with hinged bars under a punctual load Scale = 1/10
01-0031SSLLB_FEM
Units I. S.
Geometry Elements Length (m) Area (m2)
AC 0.5 2 2 x 10-4 CB 0.5 2 2 x 10-4 CD 2.5 1 x 10-4 BD 2 1 x 10-4
Materials properties Longitudinal elastic modulus: E = 1.962 x 1011 Pa.
Boundary conditions
Outer: Hinge at A and B,
Inner: None.
Test 01-0031SSLLB_FEM
97
Loading
External: Punctual force at D: Fy = F = - 9.81 x 103 N.
Internal: None.
2.31.3 Displacements at C and D
Reference solution Displacement method.
Finite elements modeling
Linear element: beam,
4 nodes,
4 linear elements.
Displacements shape Truss with hinged bars under a punctual load Scale = 1/9
Deformed
2.31.4 Results sheet Results comparison: horizontal displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In C mm 0.26517 0.26469 -0.18% CM2 In D mm 3.47902 3.47531 -0.11%
Results comparison: vertical displacement Solver Positioning Units Reference AD 2010 Deviation CM2 In C mm 0.08839 0.08817 -0.25% CM2 In D mm -5.60084 -5.5950 -0.10%
Test 01-0032SSLLB_FEM
98
2.32 Test No. 01-0032SSLLB_FEM: Beam on elastic soil, free ends
2.32.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 15/89;
Analysis type: static linear (plane problem);
Element type: linear.
2.32.2 Overview A beam under 3 punctual loads lays on a soil of constant linear stiffness.
Beam on elastic soil, free ends Scale = 1/21 01-0032SSLLB_FEM
Units I. S.
Geometry
L = (π 10 )/2,
I = 10-4 m4.
Materials properties Longitudinal elastic modulus: E = 2.1 x 1011 Pa.
Boundary conditions
Outer: o Free A and B extremities, o Constant linear stiffness of soil ky = K = 840000 N/m2.
Inner: None.
Test 01-0032SSLLB_FEM
99
Loading
External: Punctual load at A, C and B: Fy = F = - 10000 N.
Internal: None.
2.32.3 Bending moment and displacement at C
Reference solution
β = 4
K/(4EI)
ϕ = βL/2
λ = sh (2ϕ) + sin (2ϕ)
Bending moment: MC = (F/(4β))(ch(2ϕ) - cos (2ϕ) – 8sh(ϕ)sin(ϕ))/λ
Vertical displacement: vC = - (Fβ/(2K))( ch(2ϕ) + cos (2ϕ) + 8ch(ϕ)cos(ϕ) + 2)/λ
Finite elements modeling
Linear element: beam,
72 nodes,
71 linear elements.
Bending moment diagram
Beam on elastic soil, free ends Scale = 1/20 Bending moment
Test 01-0032SSLLB_FEM
100
2.32.4 Displacements at A
Reference solution
Vertical displacement: vA = (2Fβ/K)( ch(ϕ)cos(ϕ) + ch(2ϕ) + cos(2ϕ))/λ
Rotation about z-axis
θA = (-2Fβ2/K)( sh(ϕ)cos(ϕ) - sin(ϕ)ch(ϕ) + sh(2ϕ) - sin(2ϕ))/λ
Finite elements modeling
Linear element: beam,
72 nodes,
71 linear elements
2.32.5 Results sheet
1 Results comparison: bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 In C Nm 5759 5772.05 0.23%
2 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In C m -0.006844 -0.006844 0.00%
3 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In A m -0.007854 -0.00786073 0.09%
4 Results comparison: rotation θ about z-axis
Solver Positioning Units Reference AD 2010 Deviation CM2 In A rad -0.000706 -0.000707 0.14%
Test 01-0033SFLLA_FEM
101
2.33 Test No. 01-0033SFLLA_FEM: EDF Pylon
2.33.1 Description sheet
Reference: Internal GRAITEC test;
Analysis type: static linear, Eulerian buckling;
Element type: linear
2.33.2 Overview
Units
I. S.
Geometry
Test 01-0033SFLLA_FEM
102
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: o Hinged support, o For the modeling, a fixed restraint and 4 beams were added at the pylon supports level.
Inner: None.
Loading
External: Punctual loads corresponding to a wind load.
o FX = 165550 N, FY = - 1240 N, FZ = - 58720 N on the main arms ?, o FX = 50250 N, FY = - 1080 N, FZ = - 12780 N on the upper arm, o FX = 11760 N, FY = 0 N, FZ = 0 N on the lower horizontal frames
Internal: None.
Test 01-0033SFLLA_FEM
103
2.33.3 Displacement of the model in the linear elastic range
Reference solution
Software ANSYS 5.3 NE/NASTRAN 7.0 Max deflection (m) 0.714 0.714 λ dominating mode 2.77 2.77
Finite elements modeling
Linear element: beam, imposed mesh,
402 nodes,
1034 elements.
Test 01-0033SFLLA_FEM
104
Deformed shape
Buckling modal deformation (dominating mode)
Test 01-0033SFLLA_FEM
105
2.33.4 Results sheet
1 Results comparison: displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 Top of pylon m 0.714 0.71254 -0.20%
2 Results comparison: dominating buckling mode
Solver Mode Units Reference AD 2010 Deviation CM2 λcritique - 2.77 2.830 2.17%
Test 01-0034SSLLB_FEM
106
2.34 Test No. 01-0034SSLLB_FEM: Beam on elastic soil, hinged ends
2.34.1 Description sheet Reference: Structure Calculation Software Validation Guide, test SSLL 16/89;
Analysis type: static linear (plane problem);
Element type: linear.
2.34.2 Overview A beam under a punctual load, a distributed load and two torques lays on a soil of constant linear stiffness.
Beam on elastic soil, hinged ends Scale = 1/27 01-0034SSLLB_FEM
Units I. S.
Geometry
L = (π 10 )/2,
I = 10-4 m4.
Materials properties Longitudinal elastic modulus: E = 2.1 x 1011 Pa.
Boundary conditions
Outer: o Free A and B ends, o Soil with a constant linear stiffness ky = K = 840000 N/m2.
Inner: None.
Test 01-0034SSLLB_FEM
107
Loading
External: o Punctual force at D: Fy = F = - 10000 N, o Uniformly distributed force from A to B: fy = p = - 5000 N/m, o Torque at A: Cz = -C = -15000 Nm, o Torque at B: Cz = C = 15000 Nm.
Internal: None.
2.34.3 Displacement and support reaction at A
Reference solution
β = 4
K/(4EI)
ϕ = βL/2
λ = ch(2ϕ) + cos(2ϕ)
Vertical support reaction:
VA = -p(sh(2ϕ) + sin(2ϕ)) - 2βFch(ϕ)cos(ϕ) + 2β2C(sh(2ϕ) - sin(2ϕ)) x 1
2βλ
Rotation about z-axis:
θA = p(sh(2ϕ) – sin(2ϕ)) + 2βFsh(ϕ)sin(ϕ) - 2β2C(sh(2ϕ) + sin(2ϕ)) x 1
(K/β)λ
Finite elements modeling
Linear element: beam,
50 nodes,
49 linear elements.
Deformed shape Beam on elastic soil, hinged ends Scale = 1/20
Deformed
Test 01-0034SSLLB_FEM
108
2.34.4 Displacement and bending moment at D
Reference solution
Vertical displacement:
vD = 2p(λ - 2ch(ϕ)cos(ϕ)) + βF(sh(2ϕ) – sin(2ϕ)) - 8β2Csh(ϕ)sin(ϕ) x 1
2Kλ
Bending moment:
MD = 4psh(ϕ)sin(ϕ) + βF(sh(2ϕ) + sin(2ϕ)) - 8β2Cch(ϕ)cos(ϕ) x 1
4β2λ
Finite elements modeling
Linear element: beam,
50 nodes,
49 linear elements.
Bending moment diagram
Beam on elastic soil, hinged ends Scale = 1/20 Bending moment
Test 01-0034SSLLB_FEM
109
2.34.5 Results sheet
1 Results comparison: rotation around z
Solver Positioning Units Reference AD 2010 Deviation CM2 In A rad 0.003045 0.003043 -0.07%
2 Results comparison: vertical reaction
Solver Positioning Units Reference AD 2010 Deviation CM2 In A N -11674 -11644.36 -0.25%
3 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In D cm -0.423326 -0.42330 -0.01%
4 Results comparison: bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 In D Nm -33840 -33835.87 -0.01%
Test 01-0035SSLPB_FEM
110
2.35 Test No. 01-0035SSLPB_FEM: Plate with in plane bending and shear
2.35.1 Description sheet Reference: Structure Calculation Software Validation Guide, test SSLP 01/89;
Analysis type: static linear (plane problem);
Element type: planar.
CAD tolerance 0.1 mm
2.35.2 Overview
Units
I. S.
Geometry
Thickness: h = 1 mm,
Length: L = 48 mm,
Height: H = 12 mm.
Materials properties
Longitudinal elastic modulus: E = 3 x 1010 Pa,
Poisson's ratio: ν = 0.25.
Boundary conditions
Outer: fixed in any point on the edge x = 0.
Inner: None.
Loading
External: Uniformly distributed force at any point x = 48 mm: fy = p = - 3333.33 N/m.
Internal: None.
2.35.3 Planes stresses in (x,y)
Reference solution
Analytical method by Airy function:
σxx = 12Py(x - L)
H3
σyy = 0
σxy = 6P(
H2
4 - y2)
H3
Test 01-0035SSLPB_FEM
111
Finite elements modeling
Planar element: shell,
784 nodes,
720 planar elements.
2.35.4 Results sheet
1 Results comparison: σxx plane stress
Solver Positioning Units Reference AD 2010 Deviation CM2 At (0, H/2) MPa -80 -79.46 -0.68%
2 Results comparison: σxx plane stress
Solver Positioning Units Reference AD 2010 Deviation CM2 At (0, -H/2) MPa -80 -79.46 -0.68%
3 Results comparison: σxy plane stress
Solver Positioning Units Reference AD 2010 Deviation CM2 At any point y = 0 MPa -5 -4.97 -0.60%
Test 01-0036SSLSB_FEM
112
2.36 Test No. 01-0036SSLSB_FEM: Simply supported square plate
2.36.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 02/89;
Analysis type: static linear;
Element type: planar.
2.36.2 Overview A plate simply supported on its perimeter and loaded with its self weight only.
Simply supported square plate Scale = 1/9 01-0036SSLSB_FEM
Units I. S.
Geometry
Side = 1 m,
Thickness h = 0.01m.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7950 kg/m3.
Boundary conditions
Outer: o Simple support on the plate perimeter, o For the modeling, we add a fixed support at B.
Inner: None.
Test 01-0036SSLSB_FEM
113
Loading
External: Self weight (gravity = 9.81 m/s2).
Internal: None.
2.36.3 Vertical displacement at O
Reference solution
According to Love- Kirchhoff hypothesis, the displacement w at a point (x,y):
w(x,y) = Σ wmnsinmπxsinnπy
where wmn = 192ρg(1 - ν2)
mn(m2 + n2)π6Eh2
Finite elements modeling
Planar element: shell,
441 nodes,
400 planar elements.
Deformed shape
Simply supported square plate Scale = 1/6 Deformed
2.36.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In O μm -0.158 -0.16491 4.37%
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114
2.37 Test No. 01-0037SSLSB_FEM: Caisson beam in torsion
2.37.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 05/89;
Analysis type: static linear;
Element type: planar.
2.37.2 Overview A caisson beam fixed at one end is loaded with torsion.
Caisson beam in torsion Scale = 1/4 01-0037SSLSB_FEM
Units I. S.
Geometry
Length; L = 1m,
Square section of side: b = 0.1 m,
Thickness = 0.005 m.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Beam fixed at end x = 0;
Inner: None.
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115
Loading
External: Torsion moment M = 10N.m applied to the free end (for modeling, 4 forces of 50 N).
Internal: None.
2.37.3 Displacement and stress at two points
Reference solution
The reference solution is determined by averaging the results of several calculation software with implemented finite elements method.
Points coordinates:
A (0,0.05,0.5)
B (-0.05,0,0.8)
Note: point O is the origin of the coordinate system (x,y,z).
Finite elements modeling
Planar element: shell,
90 nodes,
88 planar elements.
Deformed shape
Caisson beam in torsion Scale = 1/4 Deformed
Test 01-0037SSLSB_FEM
116
2.37.4 Results sheet
1 Results comparison: displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 In A m -0.617 x 10-6 -0.6159 x 10-6 -0.18% CM2 In B m -0.987 x 10-6 -0.9868 x 10-6 -0.02%
2 Results comparison: rotation about Z-axis
Solver Positioning Units Reference AD 2010 Deviation CM2 In A rad 0.123 x 10-4 0.123 x 10-4 0.00% CM2 In B rad 0.197 x 10-4 0.197 x 10-4 0.00%
3 Results comparison: σxy stress
Solver Positioning Units Reference AD 2010 Deviation CM2 In A MPa -0.11 -0.10 -9.09% CM2 In B MPa -0.11 -0.10 -9.09%
Test 01-0038SSLSB_FEM
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2.38 Test No. 01-0038SSLSB_FEM: Thin cylinder under a uniform radial pressure
2.38.1 Description sheet Reference: Structure Calculation Software Validation Guide, test SSLS 06/89;
Analysis type: static elastic;
Element type: planar.
2.38.2 Overview A cylinder of length L and radius R loaded with an uniform internal pressure.
Thin cylinder under a uniform radial pressure Scale = 1/18 01-0038SSLSB_FEM
Units I. S.
Geometry
Length: L = 4 m,
Radius: R = 1 m,
Thickness: h = 0.02 m.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: o Free conditions o For the modeling, only ¼ of the cylinder is considered and the symmetry conditions are applied. On the
other side, we restrained the displacements at a few nodes in order to make the model stable.
Inner: None.
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118
Loading
External: Uniform internal pressure: p = 10000 Pa,
Internal: None.
2.38.3 Stresses in all points
Reference solution
Stresses in the planar elements coordinate system (x axis is parallel with the length of the cylinder):
σxx = 0
σyy = pRh
Finite elements modeling
Planar element: shell,
209 nodes,
180 planar elements.
2.38.4 Cylinder deformation in all points
Radial deformation:
δR = pR2
Eh
Longitudinal deformation:
δL = -pRνL
Eh
2.38.5 Results sheet
1 Results comparison: σxx stress
Solver Positioning Units Reference AD 2010 Deviation CM2 At all points Pa 0 0 -
2 Results comparison: σyy stress
Solver Positioning Units Reference AD 2010 Deviation CM2 At all points Pa 5 x 105 5 x 105 0.00%
3 Results comparison: δL radial deformation of the cylinder
Solver Positioning Units Reference AD 2010 Deviation CM2 At all points m 2.38 x 10-6 2.39 x 10-6 0.42%
4 Results comparison: δL longitudinal deformation of the cylinder
Solver Positioning Units Reference AD 2010 Deviation CM2 At all points m -2.86 x 10-6 -2.85 x 10-6 0.34%
Test 01-0039SSLSB_FEM
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2.39 Test No. 01-0039SSLSB_FEM: Square plate under planar stresses
2.39.1 Description sheet Reference: Internal GRAITEC test;
Analysis type: static linear;
Element type: planar (membrane).
2.39.2 Overview A square plate of 2 x 2 m is fixed on 3 sides and has a surface load "p" on it’s upper face.
Square plate under planar stresses Scale = 1/19 Modeling
[ ]1;1, −∈ηξ
Units I. S.
Geometry
Thickness: e = 1 m,
4 square elements of side h = 1 m.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Fixed on 3 sides,
Inner: None.
Loading
External: Uniform load p = -1. 108 N/ml on the upper surface,
Internal: None.
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120
2.39.3 Displacement of the model in the linear elastic range Reference solution The reference displacements are calculated on nodes 7 and 9.
v9 = -6ph(3 + ν)(1 - ν2)
E(8(3 - ν)2 - (3 + ν)2) = -0.1809 x 10-3 m,
v7 = 4(3 - ν)3 + ν v9 = -0.592 x 10-3 m,
For element 1.4: (For the stresses calculated above, the abscissa point (x = 0; y = 0) corresponds to node 8.)
σyy = E
1 - ν2 (v9 - v7)
2h (1 + ξ) for
σxx = νσyy for
σxy = E
1 + ν (v9 + v7) + η(v9 - v7)
4h (1 + ξ) for
Finite elements modeling
Planar element: membrane, imposed mesh,
9 nodes,
4 surface quadrangles.
Deformed shape
ξ = -1 ; σxx = 0 ξ = 0 ; σxx = -14.23 MPa ξ = 1 ; σxx = -28.46 MPa
η = -1 ; ξ = 0 ; σxy = -47.82 MPa η = 0 ; ξ = 0 ; σxy = -31.21 MPa η= 1 ; ξ = 0 ; σxy = -14.61 MPa
ξ = -1 ; σyy = 0 ξ = 0 ; σyy = -47.44 MPa ξ = 1 ; σyy = -94.88 MPa
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2.39.4 Results sheet
1 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 Element 1.4 node 7 mm -0.592 -0.592 0.00% CM2 Element 1.4 node 9 mm -0.1809 -0.1809 0.00%
2 Results comparison: σxx stresses
Solver Positioning Units Reference AD 2010 Deviation CM2 Element 1.4 in x = 0 m MPa 0 0 - CM2 Element 1.4 in x = 0.5 m MPa -14.23 -14.23 0.00% CM2 Element 1.4 in x = 1 m MPa -28.46 -28.46 0.00%
3 Results comparison: σyy stresses
Solver Positioning Units Reference AD 2010 Deviation CM2 Element 1.4 in x = 0 m MPa 0 0 - CM2 Element 1.4 in x = 0.5 m MPa -47.44 -47.44 0.00% CM2 Element 1.4 in x = 1 m MPa -94.88 -94.88 0.00%
4 Results comparison: σxy stresses
Solver Positioning Units Reference AD 2010 Deviation CM2 Element 1.4 in x = 0 m MPa -14.66 -14.61 -0.34% CM2 Element 1.4 in x = 0.5 m MPa -31.21 -31.21 0.00% CM2 Element 1.4 in x = 1 m MPa -47.82 -47.82 0.00%
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2.40 Test No. 01-0040SSLSB_FEM: Stiffen membrane
2.40.1 Description sheet
Reference: Klaus-Jürgen Bathe - Finite Element Procedures in Engineering Analysis, Example 5.13;
Analysis type: static linear;
Element type: planar (membrane).
2.40.2 Overview The 8 x 12 cm plate is fixed in the middle on 3 supports and has a " punctual load "P at its free node A.
[ ]1;1, −∈ηξ
Units I. S.
Geometry
Thickness: e = 0.1 cm,
Length: l = 8 cm,
Width: B = 12 cm.
Materials properties
Longitudinal elastic modulus: E = 30 x 106 N/cm2,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Fixed on 3 sides,
Inner: None.
Loading
External: Uniform load Fx = F = 6000 N at A,
Internal: None.
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123
2.40.3 Results of the model in the linear elastic range
Reference solution
Point B is the origin of the coordinate system used for the results positions.
( ) ( )
( )
( )⎪⎩
⎪⎨
⎧
−=−==−=−==
=−=+
+−=
⎪⎩
⎪⎨
⎧
==−====
===
⎪⎩
⎪⎨
⎧
==−====
==−
−=
=+
=+⎟⎟
⎠
⎞⎜⎜⎝
⎛+
+−
== −
MPa 96.17N/cm 1796 ;1MPa 98.8N/cm 898 ;0
0 ;1for 1
81
MPa 55.11N/cm 1155 ;1MPa 77.5N/cm 577 ;0
0 ;1for
MPa 49.38N/cm 3849 ;1MPa 24.19N/cm 1924 ;0
0 ;1for 1
21
3410.97510.367410.2
6000
211
12
3
2xy1
2xy1
xy1
1
2yy1
2yy1
yy1
11
2xx1
2xx1
xx1
21
466
222
σξσξ
σξξ
νσ
σηση
σηνσσ
σηση
σηη
νσ
νν
buE
auE
cm
aES
baeabE
FKFu
Axy
xxyy
Axx
A
Finite elements modeling
Planar element: membrane, imposed mesh,
6 nodes,
2 quadrangle planar elements and 1 bar.
Deformed shape
Test 01-0040SSLSB_FEM
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2.40.4 Results sheet
1 Results comparison: horizontal displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 Element 1 in A cm -9.34 10-4 -9.34 x 10-4 0.00%
2 Results comparison: σxx stress
Solver Positioning Units Reference AD 2010 Deviation CM2 Element 1 in y = 0 cm MPa 38.49 38.49 0.00% CM2 Element 1 in y = 3 cm MPa 19.24 19.24 0.00% CM2 Element 1 in y = 6 cm MPa 0 0 -
3 Results comparison: σyy stress
Solver Positioning Units Reference AD 2010 Deviation CM2 Element 1 in y = 0 cm MPa 11.55 11.55 0.00% CM2 Element 1 in y = 3 cm MPa 5.77 5.77 0.00% CM2 Element 1 in y = 6 cm MPa 0 0 -
4 Results comparison: σxy stress
Solver Positioning Units Reference AD 2010 Deviation CM2 Element 1 in x = 0 cm MPa 0 0 - CM2 Element 1 in x = 4 cm MPa -8.98 -8.98 0.00% CM2 Element 1 in x = 8 cm MPa -17.96 -17.96 0.00%
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2.41 Test No. 01-0041SSLLB_FEM: Beam on two supports considering the shear force 2.41.1 Description sheet
Reference: Internal GRAITEC test;
Analysis type: static linear (plane problem);
Element type: linear.
2.41.2 Overview The 300 cm long beam consists of an I shaped profile of a total height of 20.04 cm, a 0.96 cm thick web and 20.04 cm wide flanges of 1.46 cm thick.
Units I. S.
Geometry l = 300 cm h = 20.04 cm b= 20.04 cm tw = 1.46 cm tf = 0.96 cm Sx= 74.95 cm2 Iz = 5462 cm4 Sy = 16.43 cm2
Materials properties
Longitudinal elastic modulus: E = 2285938 daN/cm2,
Transverse elastic modulus G = 879207 daN/cm2
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: o Simple support on node 11, o For the modeling, put an hinge at node 1 (instead of a simple support).
Inner: None.
Test 01-0041SSLLB_FEM
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Loading
External: Vertical punctual load P = -20246 daN at node 6,
Internal: None.
2.41.3 Vertical displacement of the model in the linear elastic range
Reference solution
The reference displacement is calculated in the middle of the beam, at node 6.
( )cm 017.1105.0912.0
43.163.012
22859384
300202465462228593848
30020246448
33
6 −=−−=
+
−+
−=+=
xx
xxx
xGSPl
EIPlv
shear
y
flexion
z
876876
Finite elements modeling
Planar element: S beam, imposed mesh,
11 nodes,
10 linear elements.
Deformed shape
2.41.4 Results sheet
Results comparison: vertical displacement
Solver Model Units Reference AD 2010 DeviationCM2 At node 6 cm -1.017 -1.017 0.00%
Test 01-0042SSLSB_FEM
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2.42 Test No. 01-0042SSLSB_FEM: Thin cylinder under a uniform axial load 2.42.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 07/89;
Analysis type: static elastic;
Element type: planar.
2.42.2 Overview A cylinder of radius R and length L under a uniform axial load.
Thin cylinder under a uniform axial load Scale = 1/19 01-0042SSLSB_FEM
Units I. S.
Geometry
Thickness: h = 0.02 m,
Length: L = 4 m,
Radius: R = 1 m.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: o Null axial displacement at the left end: vz = 0, o For the modeling, only a ¼ of the cylinder is considered.
Inner: None.
Loading
External: Uniform axial load q = 10000 N/m
Inner: None.
Test 01-0042SSLSB_FEM
128
2.42.3 Stress in all points
Reference solution x axis of the local coordinate system of planar elements is parallel to the cylinders axis.
σxx = qh
σyy = 0
Finite elements modeling
Planar element: shell, imposed mesh,
697 nodes,
640 surface quadrangles.
2.42.4 Cylinder deformation at the free end
Reference solution
δL longitudinal deformation of the cylinder:
δL = qLEh
δR radial deformation of the cylinder:
δR = -qνREh
Finite elements modeling
Planar element: shell, imposed mesh,
697 nodes,
640 surface quadrangles.
Deformation shape Thin cylinder under a uniform axial load Scale = 1/22 Deformation shape
Test 01-0042SSLSB_FEM
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2.42.5 Results sheet
1 Results comparison: σxx stress
Solver Positioning Units Reference AD 2010 Deviation CM2 At all points Pa 5 x 105 5 x 105 0.00%
2 Results comparison: σyy stress
Solver Positioning Units Reference AD 2010 Deviation CM2 At all points Pa 0 0 -
3 Results comparison: δL longitudinal deformation
Solver Positioning Units Reference AD 2010 Deviation CM2 At the free end m 9.52 x 10-6 9.52 x 10-6 0.00%
4 Results comparison: δR radial deformation
Solver Positioning Units Reference AD 2010 Deviation CM2 At the free end m -7.14 x 10-7 -7.109 x 10-7 -0.43%
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2.43 Test No. 01-0043SSLSB_FEM: Thin cylinder under a hydrostatic pressure
2.43.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 08/89;
Analysis type: static, linear elastic;
Element type: planar.
2.43.2 Overview A cylinder of radius R and length L under a hydrostatic pressure.
Thin cylinder under a hydrostatic pressure Scale = 1/25 01-0043SSLSB_FEM
Units I. S.
Geometry
Thickness: h = 0.02 m,
Length: L = 4 m,
Radius: R = 1 m.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: For the modeling, we consider only a quarter of the cylinder, so we impose the symmetry conditions on the nodes that are parallel with the cylinder’s axis.
Inner: None.
Loading
External: Radial internal pressure varies linearly with the "p" height, p = p0 zL ,
Internal: None.
Test 01-0043SSLSB_FEM
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2.43.3 Stresses
Reference solution
x axis of the local coordinate system of planar elements is parallel to the cylinders axis.
σxx = 0
σyy = p0RzLh
Finite elements modeling
Planar element: shell, imposed mesh,
209 nodes,
180 surface quadrangles.
2.43.4 Cylinder deformation
Reference solution
δL longitudinal deformation of the cylinder:
δL = -p0Rνz2
2ELh
δL radial deformation of the cylinder:
δR = p0R2zELh
Finite elements modeling
Planar element: shell, imposed mesh,
209 nodes,
180 surface quadrangles.
Deformation shape
Thin cylinder under a hydrostatic pressure Deformed
Test 01-0043SSLSB_FEM
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2.43.5 Results sheet
1 Results comparison: σxx stress
Solver Positioning Units Reference AD 2010 Deviation CM2 At all points Pa 0 0 -
2 Results comparison: σyy stress
Solver Positioning Units Reference AD 2010 Deviation CM2 In z = L/2 Pa 5 x 105 5 x 105 0.00%
3 Results comparison: δL longitudinal deformation of the cylinder
Solver Positioning Units Reference AD 2010 Deviation CM2 Inferior extremity m -2.86 x 10-6 -2.854 x 10-6 -0.21%
4 Results comparison: δL radial deformation of the cylinder
Solver Positioning Units Reference AD 2010 Deviation CM2 In z = L/2 m 2.38 x 10-6 2.38 x 10-6 0.00%
Test 01-0044SSLSB_FEM
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2.44 Test No. 01-0044SSLSB_FEM: Thin cylinder under its self weight
2.44.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 09/89;
Analysis type: static, linear elastic;
Element type: planar.
2.44.2 Overview A cylinder of R radius and L length subject of self weight only.
Thin cylinder under its self weight Scale = 1/24 01-0044SSLSB_FEM
Units I. S.
Geometry
Thickness: h = 0.02 m,
Length: L = 4 m,
Radius: R = 1 m.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: γ = 7.85 x 104 N/m3.
Boundary conditions
Outer: o Null axial displacement at z = 0, o For the modeling, we consider only a quarter of the cylinder, so we impose the symmetry conditions on the
nodes that are parallel with the cylinder’s axis.
Inner: None.
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Loading
External: Cylinder self weight,
Internal: None.
2.44.3 Stresses
Reference solution
x axis of the local coordinate system of planar elements is parallel to the cylinders axis.
σxx = γz
σyy = 0
Finite elements modeling
Planar element: shell, imposed mesh,
697 nodes,
640 surface quadrangles.
2.44.4 Cylinder deformation
Reference solution
δL longitudinal deformation of the cylinder:
δL = γz2
2E
δR radial deformation of the cylinder:
δR = -γνRz
E
2.44.5 Results sheet
1 Results comparison: σyy stress
Solver Positioning Units Reference AD 2010 Deviation CM2 At all points Pa 0 0 -
Note: To obtain this result, you must generate a calculation note “Planar elements stresses by load case in neutral fiber" with results on center.
2 Results comparison: σxx stress
Solver Model Units Reference AD 2010 Deviation CM2 for z = L Pa 3.14 x 105 3.11 x 105 -0.95%
3 Results comparison: δL longitudinal deformation
Solver Model Units Reference AD 2010 Deviation CM2 for z = L m x 10-6 2.99 2.99 0.00%
4 Results comparison: δR radial deformation
Solver Model Units Reference AD 2010 Deviation CM2 for z = L m x 10-6 -0.44 -0.44 0.00%
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2.45 Test No. 01-0045SSLSB_FEM: Torus with uniform internal pressure
2.45.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 10/89;
Analysis type: static, linear elastic;
Element type: planar.
2.45.2 Overview A torus with radius "a", and in transverse section with a radius "b", loaded with an uniform internal pressure.
Torus with uniform internal pressure
01-0045SSLSB_FEM
Units I. S.
Geometry
Thickness: h = 0.02 m,
Transverse section radius: b = 1 m,
Average radius of curvature: a = 2 m.
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: For the modeling, only 1/8 of the cylinder is considered, so the symmetry conditions are imposed to end nodes.
Inner: None.
Test 01-0045SSLSB_FEM
136
Loading
External: Uniform internal pressure p = 10000 Pa
Internal: None.
2.45.3 Stresses
Reference solution
(See stresses description on the first scheme of the overview)
If a – b ≤ r ≤ a + b
σ11 = pb2h
r + ar
σ22 = pb2h
Finite elements modeling
Planar element: shell, imposed mesh,
361 nodes,
324 surface quadrangles.
2.45.4 Cylinder deformation
Reference solution
δR radial deformation of the torus:
δR = pb
2Eh (r - ν(r + a))
Finite elements modeling
Planar element: shell, imposed mesh,
361 nodes,
324 surface quadrangles.
2.45.5 Results sheet
1 Results comparison: σ11 stresses
Solver Positioning Units Reference AD 2010 Deviation CM2 for r = a - b Pa 7.5 x 105 7.43 x 105 -0.94% CM2 for r = a + b Pa 4.17 x 105 4.15 x 105 -0.48%
2 Results comparison: σ22 stress
Solver Positioning Units Reference AD 2010 Deviation CM2 for all r Pa 2.50 x 105 2.49 x 105
3 Results comparison: δL radial deformations of the torus
Solver Model Units Reference AD 2010 Deviation CM2 for r = a - b m 1.19 x 10-7 1.18 x 10-7 0.84% CM2 for r = a + b m 1.79 x 10-6 1.79 x 10-6 0.00%
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2.46 Test No. 01-0046SSLSB_FEM: Spherical shell under internal pressure
2.46.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 14/89;
Analysis type: static, linear elastic;
Element type: planar.
2.46.2 Overview A spherical shell of radius R2 is subjected to an internal pressure.
Spherical shell under internal pressure 01-0046SSLSB_FEM
Test 01-0046SSLSB_FEM
138
Units I. S.
Geometry
Thickness: h = 0.02 m,
Radius: R2 = 1 m,
θ = 90° (hemisphere).
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Simple support (null displacement along vertical displacement) on the shell perimeter.
For modeling, we consider only half of the hemisphere, so we impose symmetry conditions (DOF restrains placed in the vertical plane xy in translation along z and in rotation along x and y). In addition, the node at the top of the shell is restrained in translation along x to assure the stability of the structure during calculation).
Inner: None.
Loading
External: Uniform internal pressure p = 10000 Pa
Internal: None.
2.46.3 Stresses
Reference solution (See stresses description on the first scheme of the overview)
If 0° ≤ θ ≤ 90°
σ11 = σ22 = pR2
2
2h
Finite elements modeling
Planar element: shell, imposed mesh,
343 nodes,
324 planar elements.
2.46.4 Cylinder deformation
Reference solution
δR radial deformation of the calotte:
δR = pR2
2 (1 - ν) sin θ 2Eh
Finite elements modeling
Planar element: shell, imposed mesh,
343 nodes,
324 planar elements.
Test 01-0046SSLSB_FEM
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Deformed shape
Spherical shell under internal pressure Scale = 1/11 Deformed
2.46.5 Results sheet
1 Results comparison: σ11 stress
Solver Positioning Units Reference AD 2010 Deviation CM2 for all θ Pa 2.50 x 105 2.50 x 105 0.00%
2 Results comparison: σ22 stress
Solver Positioning Units Reference AD 2010 Deviation CM2 for all θ Pa 2.50 x 105 2.50 x 105 0.00%
2 Results comparison: δR radial deformations
Solver Model Units Reference AD 2010 Deviation CM2 for θ = 90° m 8.33 x 10-7 8.34 x 10-7 0.12%
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2.47 Test No. 01-0047SSLSB_FEM: Spherical shell under its self weight
2.47.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 17/89;
Analysis type: static, linear elastic;
Element type: planar.
CAD tolerance 0.5 mm
2.47.2 Overview A spherical shell of radius R2 is subjected to its own weight.
Spherical shell under its self weight 01-0047SSLSB_FEM
Test 01-0047SSLSB_FEM
141
Units I. S.
Geometry
Thickness: h = 0.02 m,
Radius: R2 = 1 m,
θ = 90° (hemisphere).
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: γ = 7.85 x 104 N/m3.
Boundary conditions
Outer: Simple support (null displacement along vertical displacement) on the shell perimeter.
For modeling, we consider only a quarter of the hemisphere, so we impose symmetry conditions (DOF restrains placed in the vertical yz plane in translation along x and in rotation along y and z. Nodes placed on the vertical xy plane are restrained in translation along z and in rotation along x and y),
Inner: None.
Loading
External: Self weight, the vertical axis is y-axis,
Internal: None.
2.47.3 Stresses
Reference solution (See stresses description on the first scheme of the overview)
σ11 = γR2
1 + cosθ
σ22 = -γR2 ( 1
1 + cosθ - cosθ)
Finite elements modeling
Planar element: shell, imposed mesh,
2071 nodes,
2025 planar elements.
CAD tolerance. = 0.0005 m
2.47.4 Cylinder radial deformation
Reference solution
δR = -γR2
2 sinθ E (
1 + ν1 + cosθ - cosθ)
Finite elements modeling
Planar element: shell, imposed mesh,
2071 nodes,
2025 planar elements.
Test 01-0047SSLSB_FEM
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2.47.5 Results sheet
1 Results comparison: σ11 stress
Solver Positioning Units Reference AD 2010 Deviation CM2 for all θ = 90° Pa 7.85 x 104 7.83 x 104 -0.25%
2 Results comparison: σ22 stress
Solver Positioning Units Reference AD 2010 Deviation CM2 for all θ = 90° Pa -7.85 x 104 -7.90 x 104 -0.63%
3 Results comparison: δR radial deformation
Solver Positioning Units Reference AD 2010 Deviation CM2 for all θ = 90° m 4.86 x 10-7 4.85 x 10-7 -0.20%
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2.48 Test No. 01-0048SSLSB_FEM: Pinch cylindrical shell
2.48.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 20/89;
Analysis type: static, linear elastic;
Element type: planar.
2.48.2 Overview A cylinder of length L is pinched by 2 diametrically opposite forces (F).
Pinch cylindrical shell 01-0048SSLSB_FEM
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144
Units
I. S.
Geometry
Length: L = 10.35 m (total length),
Radius: R = 4.953 m,
Thickness: h = 0.094 m.
Materials properties
Longitudinal elastic modulus: E = 10.5 x 106 Pa,
Poisson's ratio: ν = 0.3125.
Boundary conditions
Outer: For the modeling, we consider only half of the cylinder, so we impose symmetry conditions (nodes in the horizontal xz plane are restrained in translation along y and in rotation along x and z),
Inner: None.
Loading
External: 2 punctual loads F = 100 N,
Internal: None.
2.48.3 Vertical displacement at point A
Reference solution
The reference solution is determined by averaging the results of several calculation software with implemented finite elements method. 2% uncertainty about the reference solution.
Finite elements modeling
Planar element: shell, imposed mesh,
777 nodes,
720 surface quadrangles.
2.48.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference Reference margin AD 2010 Deviation CM2 At point A m -113.9 x 10-3 ± 2% -113.29 x 10-3 -0.53%
Test 01-0049SSLSB_FEM
145
2.49 Test No. 01-0049SSLSB_FEM: Spherical shell with holes
2.49.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 21/89;
Analysis type: static, linear elastic;
Element type: planar.
2.49.2 Overview A spherical shell with holes is subjected to 4 forces, opposite 2 by 2.
Spherical shell with holes
01-0049SSLSB_FEM
Units
I. S.
Geometry
Radius: R = 10 m Thickness: h = 0.04 m, Opening angle of the hole: ϕ0 = 18°.
Materials properties
Longitudinal elastic modulus: E = 6.285 x 107 Pa, Poisson's ratio: ν = 0.3.
Test 01-0049SSLSB_FEM
146
Boundary conditions
Outer: For modeling, we consider only a quarter of the shell, so we impose symmetry conditions (nodes in the vertical yz plane are restrained in translation along x and in rotation along y and z. Nodes on the vertical xy plane are restrained in translation along z and in rotation along x and y),
Inner: None.
Loading
External: Punctual loads F = 1 N, according to the diagram,
Internal: None.
2.49.3 Horizontal displacement at point A
Reference solution The reference solution is determined by averaging the results of several calculation software with implemented finite elements method. 2% uncertainty about the reference solution.
Finite elements modeling
Planar element: shell, imposed mesh,
99 nodes,
80 surface quadrangles.
Deformed shape Spherical shell with holes Scale = 1/79
Deformed
2.49.4 Results sheet
Results comparison: horizontal displacement Solver Positioning Units Reference Reference margin AD 2010 Deviation CM2 At point A(R,0,0) mm 94.0 ± 2% 92.13 -1.99%
Test 01-0050SSLSB_FEM
147
2.50 Test No. 01-0050SSLSB_FEM: Spherical dome under a uniform external pressure
2.50.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 22/89;
Analysis type: static, linear elastic;
Element type: planar.
2.50.2 Overview A spherical dome of radius (a) is subjected to an uniform external pressure.
Spherical dome under a uniform external pressure
01-0050SSLSB_FEM
Units
I. S.
Geometry
Radius: a = 2.54 m,
Thickness: h = 0.0127 m,
Angle: θ = 75°.
Test 01-0050SSLSB_FEM
148
Materials properties
Longitudinal elastic modulus: E = 6.897 x 1010 Pa,
Poisson's ratio: ν = 0.2.
Boundary conditions
Outer: Fixed on the dome perimeter,
Inner: None.
Loading
External: Uniform pressure p = 0.6897 x 106 Pa,
Internal: None.
2.50.3 Horizontal displacement and exterior meridian stress
Reference solution
The reference solution is determined by averaging the results of several calculation software with implemented finite elements method. 2% uncertainty about the reference solution.
Finite elements modeling
Planar element: shell, imposed mesh,
401 nodes,
400 planar elements.
Deformed shape
Test 01-0050SSLSB_FEM
149
2.50.4 Results sheet
1 Results comparison: horizontal displacements
Solver Positioning Units Reference Reference margin AD 2010 DeviationCM2 In ψ = 15° m 1.73 x 10-3 ± 2% 1.7306 x 10-3 0.04% CM2 In ψ = 45° m -1.02 x 10-3 ± 2% -1.01 x 10-3
2 Results comparison: σXX external meridian stresses
Solver Model Units Reference Reference margin AD 2010 Deviation CM2 In ψ = 15° MPa -74 ± 2% -72.26 -2.35% CM2 In ψ = 45° MPa -68 ± 2% -68.99
(7.5° between two nodes)
Test 01-0051SSLSB_FEM
150
2.51 Test No. 01-0051SSLSB_FEM: Simply supported square plate under a uniform load
2.51.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 24/89;
Analysis type: static, linear elastic;
Element type: planar.
2.51.2 Overview A square plate simply supported is subjected to an uniform load.
Simply supported square plate under a uniform load Scale = 1/9 01-0051SSLSB_FEM
Units
I. S.
Geometry
Side: a =b = 1 m,
Thickness: h = 0.01 m,
Materials properties
Longitudinal elastic modulus: E = 1.0 x 107 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Simple support on the plate perimeter (null displacement along z-axis),
Inner: None
Test 01-0051SSLSB_FEM
151
Loading
External: Normal pressure of plate p pZ = -1.0 Pa,
Internal: None.
2.51.3 Vertical displacement and bending moment at the center of the plate
Reference solution
Love-Kirchhoff thin plates theory.
Finite elements modeling
Planar element: plate, imposed mesh,
361 nodes,
324 planar elements.
2.51.4 Results sheet
1 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At plate center m -4.43 x 10-3 -4.358 x 10-3 -1.61%
2 Results comparison: MX bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 At plate center Nm 0.0479 0.0471 -1.67%
3 Results comparison: MY bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 At plate center Nm 0.0479 0.0471 -1.67%
Test 01-0052SSLSB_FEM
152
2.52 Test No. 01-0052SSLSB_FEM: Simply supported rectangular plate under a uniform load
2.52.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 24/89;
Analysis type: static, linear elastic;
Element type: planar.
2.52.2 Overview A rectangular plate simply supported is subjected to an uniform load.
Simply supported rectangular plate under a uniform load Scale = 1/11 01-0052SSLSB_FEM
Units
I. S.
Geometry
Width: a = 1 m,
Length: b = 2 m,
Thickness: h = 0.01 m,
Materials properties
Longitudinal elastic modulus: E = 1.0 x 107 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Simple support on the plate perimeter (null displacement along z-axis),
Inner: None.
Test 01-0052SSLSB_FEM
153
Loading
External: Normal pressure of plate p = pZ = -1.0 Pa,
Internal: None.
2.52.3 Vertical displacement and bending moment at the center of the plate
Reference solution
Love-Kirchhoff thin plates theory.
Finite elements modeling
Planar element: plate, imposed mesh,
435 nodes,
392 surface quadrangles.
2.52.4 Results sheet
1 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At plate center m -1.1060 x 10-2 -1.1023 x 10-2 -0.33%
2 Results comparison: MX bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 At plate center Nm -0.1017 -0.1017 0.00%
3 Results comparison: MY bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 At plate center Nm -0.0464 -0.0465 0.22%
Test 01-0053SSLSB_FEM
154
2.53 Test No. 01-0053SSLSB_FEM: Simply supported rectangular plate under a uniform load
2.53.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 24/89;
Analysis type: static, linear elastic;
Element type: planar.
2.53.2 Overview A simply supported rectangular plate is subjected to an uniform load.
Simply supported rectangular plate under a uniform load Scale = 1/25 01-0053SSLSB_FEM
Units
I. S.
Geometry
Width: a = 1 m,
Length: b = 5 m,
Thickness: h = 0.01 m,
Materials properties
Longitudinal elastic modulus: E = 1.0 x 107 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Simple support on the plate perimeter (null displacement along z-axis),
Inner: None.
Test 01-0053SSLSB_FEM
155
Loading
External: Normal pressure of plate p = pZ = -1.0 Pa,
Internal: None.
2.53.3 Vertical displacement and bending moment at the center of the plate
Reference solution
Love-Kirchhoff thin plates theory.
Finite elements modeling
Planar element: plate, imposed mesh,
793 nodes,
720 surface quadrangles.
2.53.4 Results sheet
1 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At plate center m 1.416 x 10-2 1.401 x 10-2 -1.06%
2 Results comparison: MX bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 At plate center Nm 0.1246 0.1241 -0.40%
3 Results comparison: MY bending moment
Solver Positioning Units Reference AD 2010 Deviation CM2 At plate center Nm 0.0375 0.0376 0.27%
Test 01-0054SSLSB_FEM
156
2.54 Test No. 01-0054SSLSB_FEM: Simply supported rectangular plate loaded with punctual force and moments
2.54.1 Description sheet Reference: Structure Calculation Software Validation Guide, test SSLS 26/89;
Analysis type: static linear;
Element type: planar.
2.54.2 Overview A rectangular plate simply supported is subjected to a force and to punctual moments.
Simply supported rectangular plate loaded with punctual force and moments 01-0054SSLSB_FEM
Units I. S.
Geometry
Width: DA = CB = 20 m,
Length: AB = DC = 5 m,
Thickness: h = 1 m,
Materials properties
Longitudinal elastic modulus: E =1000 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Punctual support at A, B and D (null displacement along z-axis),
Inner: None.
Loading
External: o In A: MX = 20 Nm, MY = -10 Nm, o In B: MX = 20 Nm, MY = 10 Nm, o In C: FZ = -2 N, MX = -20 Nm, MY = 10 Nm, o In D: MX = -20 Nm, MY = -10 Nm,
Internal: None.
Test 01-0054SSLSB_FEM
157
2.54.3 Vertical displacement at C
Reference solution
Love-Kirchhoff thin plates theory.
Finite elements modeling
Planar element: plate, imposed mesh,
867 nodes,
800 surface quadrangles.
Deformed shape
Simply supported rectangular plate loaded with punctual force and moments Deformed
2.54.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -12.480 -12.667 1.50%
Test 01-0055SSLSB_FEM
158
2.55 Test No. 01-0055SSLSB_FEM: Shear plate perpendicular to the medium surface
2.55.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLS 27/89;
Analysis type: static;
Element type: planar.
2.55.2 Overview Rectangular plate fixed at one end, loaded with two forces.
Shear plate Scale = 1/50 01-0055SSLSB_FEM
Units
I. S.
Geometry
Length: L = 12 m,
Width: l = 1 m,
Thickness: h = 0.05 m,
Materials properties
Longitudinal elastic modulus: E =1.0 x 107 Pa,
Poisson's ratio: ν = 0.25.
Boundary conditions
Outer: Fixed AD edge,
Inner: None.
Test 01-0055SSLSB_FEM
159
Loading
External: o At B: Fz = -1.0 N, o At C: FZ = 1.0 N,
Internal: None.
2.55.3 Vertical displacement at C
Reference solution
Analytical solution.
Finite elements modeling
Planar element: plate, imposed mesh,
497 nodes,
420 surface quadrangles.
Deformed shape
Shear plate Scale = 1/35 Deformed
2.55.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference Reference margin AD 2010 Deviation CM2 At point C m 35.37 x 10-3 ± 3% 35.67 x 10-3 0.85%
Test 01-0056SSLLB_FEM
160
2.56 Test No. 01-0056SSLLB_FEM: Triangulated system with hinged bars
2.56.1 Description sheet Reference: Structure Calculation Software Validation Guide, test SSLL 12/89;
Analysis type: static (plane problem);
Element type: linear.
2.56.2 Overview A truss with hinged bars is placed on 3 punctual supports (subjected to imposed displacements) and is loaded with two 2 punctual forces. A thermal load is applied to all the bars.
Units
I. S.
Geometry
θ = 30°,
Section A1 = 1.41 x 10-3 m2,
Section A2 = 2.82 x 10-3 m2.
Materials properties
Longitudinal elastic modulus: E =2.1 x 1011 Pa,
Coefficient of linear expansion: α = 10-5 °C-1.
Boundary conditions
Outer: o Hinge at A (uA = vA = 0), o Roller supports at B and C ( uB = v’C = 0),
Inner: None.
Loading
External: o Support displacement: vA = -0.02 m ; vB = -0.03 m ; v’C = -0.015 m , o Punctual loads: FE = -150 KN ; FF = -100 KN, o Expansion effect on all bars for a temperature variation of 150° in relation with the assembly temperature
(specified geometry),
Internal: None.
Test 01-0056SSLLB_FEM
161
2.56.3 Tension force in BD bar
Reference solution Determining the hyperstatic unknown with the section cut method.
Finite elements modeling
Linear element: S beam, automatic mesh,
11 nodes,
17 S beams + 1 rigid S beam for the modeling of the simple support at C.
2.56.4 Vertical displacement at D
Reference solution vD displacement was determined by several software with implemented finite elements method.
Finite elements modeling
Linear element: S beam, automatic mesh,
11 nodes,
17 S beams + 1 rigid S beam for the modeling of simple support at C.
Deformed shape Triangulated system with hinged bars 01-0056SSLLB_FEM
2.56.5 Results sheet
1 Results comparison: FX traction force
Solver Positioning Units Reference AD 2010 Deviation CM2 BD bar N 43633 43688 0.13%
2 Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point D m -0.01618 -0.01616 -0.12%
Test 01-0057SSLSB_FEM
162
2.57 Test No. 01-0057SSLSB_FEM: 0.01m thick plate fixed on its perimeter, loaded with a uniform pressure
2.57.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLV 09/89;
Analysis type: static;
Element type: planar.
2.57.2 Overview Square plate of side "a". For the modeling, only a quarter of the plate is considered.
Units
I. S.
Geometry
Side: a = 1 m,
Thickness: h = 0.01 m,
Slenderness: λ = ah = 100.
Materials properties
Reinforcement,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Fixed sides: AB and BD,
For the modeling, we impose symmetry conditions at the CB side (restrained displacement along x and restrained rotation around y and z) and CD side (restrained displacement along y and restrained rotation around x and z),
Inner: None.
Loading
External: 1 MPa uniform pressure,
Internal: None.
Test 01-0057SSLSB_FEM
163
2.57.3 Vertical displacement at C
Reference solution
This problem has a precise analytical solution only for thin plates. Therefore we propose the solutions obtained with Serendip elements with 20 nodes or thick plate elements of 4 nodes. The expected result should be between these values at ± 5%.
Finite elements modeling
Planar element: plate, imposed mesh,
289 nodes,
256 surface quadrangles.
2.57.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -6.639 x 10-2 -6.565 x 10-2 -1.11%
Test 01-0058SSLSB_FEM
164
2.58 Test No. 01-0058SSLSB_FEM: 0.01333 m thick plate fixed on its perimeter, loaded with a uniform pressure
2.58.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLV 09/89;
Analysis type: static;
Element type: planar.
2.58.2 Overview Square plate of side "a", for the modeling, only a quarter of the plate is considered.
Units
I. S.
Geometry
Side: a = 1 m,
Thickness: h = 0.01333 m,
Slenderness: λ = ah = 75.
Materials properties
Reinforcement,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Fixed sides: AB and BD,
For the modeling, we impose symmetry conditions at the CB side (restrained displacement along x and restrained rotation around y and z) and CD side (restrained displacement along y and restrained rotation around x and z),
Inner: None.
Loading
External: 1 MPa uniform pressure,
Internal: None.
Test 01-0058SSLSB_FEM
165
2.58.3 Vertical displacement at C
Reference solution
This problem has a precise analytical solution only for thin plates. Therefore we propose the solutions obtained with Serendip elements with 20 nodes or thick plate elements of 4 nodes. The expected result should be between these values at ± 5%.
Finite elements modeling
Planar element: plate, imposed mesh,
289 nodes,
256 surface quadrangles.
2.58.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -2.8053 x 10-2 -2.7950 x 10-2 -0.37%
Test 01-0059SSLSB_FEM
166
2.59 Test No. 01-0059SSLSB_FEM: 0.02 m thick plate fixed on its perimeter, loaded with a uniform pressure
2.59.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLV 09/89;
Analysis type: static;
Element type: planar.
2.59.2 Overview Square plate of side "a", for the modeling, only a quarter of the plate is considered.
Units
I. S.
Geometry
Side: a = 1 m,
Thickness: h = 0.02 m,
Slenderness: λ = ah = 50.
Materials properties
Reinforcement,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Fixed edges: AB and BD,
For the modeling, we impose symmetry conditions at the CB side (restrained displacement along x and restrained rotation around y and z) and CD side (restrained displacement along y and restrained rotation around x and z),
Inner: None.
Loading
External: 1 MPa uniform pressure,
Internal: None.
Test 01-0059SSLSB_FEM
167
2.59.3 Vertical displacement at C
Reference solution
This problem has a precise analytical solution only for thin plates. Therefore we propose the solutions obtained with Serendip elements with 20 nodes or thick plate elements of 4 nodes. The expected result should be between these values at ± 5%.
Finite elements modeling
Planar element: plate, imposed mesh,
289 nodes,
256 surface quadrangles.
2.59.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -0.83480 x 10-2 -0.82559 x 10-2 -1.1%
Test 01-0060SSLSB_FEM
168
2.60 Test No. 01-0060SSLSB_FEM: 0.05 m thick plate fixed on its perimeter, loaded with a uniform pressure
2.60.1 Description sheet Reference: Structure Calculation Software Validation Guide, test SSLV 09/89;
Analysis type: static;
Element type: planar.
2.60.2 Overview Square plate of side "a", for the modeling, only a quarter of the plate is considered.
Units
I. S.
Geometry
Side: a = 1 m,
Thickness: h = 0.05 m,
Slenderness: λ = ah = 20.
Materials properties
Reinforcement,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Fixed edges: AB and BD,
For the modeling, we impose symmetry conditions at the CB side (restrained displacement along x and restrained rotation around y and z) and CD side (restrained displacement along y and restrained rotation around x and z),
Inner: None.
Loading
External: 1 MPa uniform pressure,
Internal: None.
Test 01-0060SSLSB_FEM
169
2.60.3 Vertical displacement at C
Reference solution
This problem has a precise analytical solution only for thin plates. Therefore we propose the solutions obtained with Serendip elements with 20 nodes or thick plate elements of 4 nodes. The expected result should be between these values at ± 5%.
Finite elements modeling
Planar element: plate, imposed mesh,
289 nodes,
256 surface quadrangles.
2.60.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -0.55474 x 10-3 -0.54987 x 10-3 -0.88%
Test 01-0061SSLSB_FEM
170
2.61 Test No. 01-0061SSLSB_FEM: 0.1 m thick plate fixed on its perimeter, loaded with a uniform pressure
2.61.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLV 09/89;
Analysis type: static;
Element type: planar.
2.61.2 Overview Square plate of side "a", for the modeling, only a quarter of the plate is considered.
Units
I. S.
Geometry
Side: a = 1 m,
Thickness: h = 0.1 m,
Slenderness: λ = ah = 10.
Materials properties
Reinforcement,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Fixed edges: AB and BD,
For the modeling, we impose symmetry conditions at the CB side (restrained displacement along x and restrained rotation around y and z) and CD side (restrained displacement along y and restrained rotation around x and z),
Inner: None.
Loading
External: 1 MPa uniform pressure,
Internal: None.
Test 01-0061SSLSB_MEF
171
2.61.3 Vertical displacement at C
Reference solution
This problem has a precise analytical solution only for thin plates. Therefore we propose the solutions obtained with Serendip elements with 20 nodes or thick plate elements of 4 nodes. The expected result should be between these values at ± 5%.
Finite elements modeling
Planar element: plate, imposed mesh,
289 nodes,
256 surface quadrangles.
2.61.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -0.78661 x 10-4 -0.78185 x 10-4 -0.61%
Test 01-0062SSLSB_MEF
172
2.62 Test No. 01-0062SSLSB_FEM: 0.01 m thick plate fixed on its perimeter, loaded with a punctual force
2.62.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLV 09/89;
Analysis type: static;
Element type: planar.
2.62.2 Overview Square plate of side "a".
0.01 m thick plate fixed on its perimeter Scale = 1/5 01-0062SSLSB_FEM
Units
I. S.
Geometry
Side: a = 1 m,
Thickness: h = 0.01 m,
Slenderness: λ = ah = 100.
Materials properties
Reinforcement,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Fixed edges,
Inner: None.
Test 01-0062SSLSB_MEF
173
Loading
External: Punctual force applied on the center of the plate: FZ = -106 N,
Internal: None.
2.62.3 Vertical displacement at point C (center of the plate)
Reference solution
This problem has a precise analytical solution only for thin plates. Therefore we propose the solutions obtained with Serendip elements with 20 nodes or thick plate elements of 4 nodes. The expected result should be between these values at ± 5%.
Finite elements modeling
Planar element: plate, imposed mesh,
961 nodes,
900 surface quadrangles.
2.62.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -0.29579 -0.292146 -1.23%
Test 01-0063SSLSB_MEF
174
2.63 Test No. 01-0063SSLSB_FEM: 0.01333 m thick plate fixed on its perimeter, loaded with a punctual force
2.63.1 Description sheet Reference: Structure Calculation Software Validation Guide, test SSLV 09/89;
Analysis type: static;
Element type: planar.
2.63.2 Overview Square plate of side a.
0.01333 m thick plate fixed on its perimeter Scale = 1/5 01-0063SSLSB_FEM
Units
I. S.
Geometry
Side: a = 1 m,
Thickness: h = 0.01333 m,
Slenderness: λ = ah = 75.
Materials properties
Reinforcement,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Fixed sides,
Inner: None.
Test 01-0063SSLSB_MEF
175
Loading
External: Punctual force applied on the center of the plate: FZ = -106 N,
Internal: None.
2.63.3 Vertical displacement at point C (the center of the plate)
Reference solution
This problem has a precise analytical solution only for thin plates. Therefore we propose the solutions obtained with Serendip elements with 20 nodes or thick plate elements of 4 nodes. The expected result should be between these values at ± 5%.
Finite elements modeling
Planar element: plate, imposed mesh,
961 nodes,
900 surface quadrangles.
2.63.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -0.12525 -0.12458 -0.53%
Test 01-0064SSLSB_MEF
176
2.64 Test No. 01-0064SSLSB_FEM: 0.02 m thick plate fixed on its perimeter, loaded with a punctual force
2.64.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLV 09/89;
Analysis type: static;
Element type: planar.
2.64.2 Overview Square plate of side "a".
0.02 m thick plate fixed on its perimeter Scale = 1/5 01-0064SSLSB_FEM
Units
I. S.
Geometry
Side: a = 1 m,
Thickness: h = 0.02 m,
Slenderness: λ = 50.
Materials properties
Reinforcement,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: Fixed edges,
Inner: None.
Test 01-0064SSLSB_MEF
177
Loading
External: punctual force applied in the center of the plate: FZ = -106 N,
Internal: None.
2.64.3 Vertical displacement at point C (the center of the plate)
Reference solution
This problem has a precise analytical solution only for thin plates. Therefore we propose the solutions obtained with Serendip elements with 20 nodes or thick plate elements of 4 nodes. The expected result should be between these values at ± 5%.
Finite elements modeling
Planar element: plate, imposed mesh,
961 nodes,
900 surface quadrangles.
2.64.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -0.037454 -0.03698 -1.26%
Test 01-0065SSLSB_MEF
178
2.65 Test No. 01-0065SSLSB_FEM: 0.05 m thick plate fixed on its perimeter, loaded with a punctual force
2.65.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLV 09/89;
Analysis type: static;
Element type: planar.
2.65.2 Overview Square plate of side a.
0.05 m thick plate fixed on its perimeter Scale = 1/5 01-0065SSLSB_FEM
Units
I. S.
Geometry
Side: a = 1 m,
Thickness: h = 0.05 m,
Slenderness: λ = 20.
Materials properties
Reinforcement,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Test 01-0065SSLSB_MEF
179
Boundary conditions
Outer: Fixed sides,
Inner: None.
Loading
External: Punctual force applied at the center of the plate: FZ = -106 N,
Internal: None.
2.65.3 Vertical displacement at point C center of the plate)
Reference solution
This problem has a precise analytical solution only for thin plates. Therefore we propose the solutions obtained with Serendip elements with 20 nodes or thick plate elements of 4 nodes. The expected result should be between these values at ± 5%.
Finite elements modeling
Planar element: plate, imposed mesh,
961 nodes,
900 surface quadrangles.
2.65.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -0.2595 x 10-2 -0.2572 x 10-2 -0.86%
Test 01-0066SSLSB_MEF
180
2.66 Test No. 01-0066SSLSB_FEM: 0.1 m thick plate fixed on its perimeter, loaded with a punctual force
2.66.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLV 09/89;
Analysis type: static;
Element type: planar.
2.66.2 Overview Square plate of side a.
0.1 m thick plate fixed on its perimeter Scale = 1/5 01-0066SSLSB_FEM
Units
I. S.
Geometry
Side: a = 1 m,
Thickness: h = 0.1 m,
Slenderness: λ = 10.
Materials properties
Reinforcement,
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3.
Test 01-0066SSLSB_MEF
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Boundary conditions
Outer: Fixed edges,
Inner: None.
Loading
External: punctual force applied in the center of the plate: FZ = -106 N,
Internal: None.
2.66.3 Vertical displacement at point C (center of the plate)
Reference solution
This problem has a precise analytical solution only for thin plates. Therefore we propose the solutions obtained with Serendip elements with 20 nodes or thick plate elements of 4 nodes. The expected result should be between these values at ± 5%.
Finite elements modeling
Planar element: plate, imposed mesh,
961 nodes,
900 surface quadrangles.
2.66.4 Results sheet
Results comparison: vertical displacement
Solver Positioning Units Reference AD 2010 Deviation CM2 At point C m -0.42995 x 10-3 -0.41209 x 10-3 -4.15
Test 01-0067SDLLB_MEF
182
2.67 Test No. 01-0067SDLLB_FEM: Vibration mode of a thin piping elbow in space (case 1) 2.67.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 14/89;
Analysis type: modal analysis (space problem);
Element type: linear.
2.67.2 Overview A thin piping elbow with a radius of 1m is fixed on its ends and loaded with its self weight only.
Vibration mode of a thin piping elbow Scale = 1/7 01-0067SDLLB_FEM
Units
I. S.
Geometry
Average radius of curvature: OA = R = 1 m,
Straight circular hollow section:
Outer diameter: de = 0.020 m,
Inner diameter: di = 0.016 m,
Section: A = 1.131 x 10-4 m2,
Flexure moment of inertia relative to the y-axis: Iy = 4.637 x 10-9 m4,
Flexure moment of inertia relative to z-axis: Iz = 4.637 x 10-9 m4,
Polar inertia: Ip = 9.274 x 10-9 m4.
Points coordinates (in m): o O ( 0 ; 0 ; 0 ) o A ( 0 ; R ; 0 ) o B ( R ; 0 ; 0 )
Test 01-0067SDLLB_MEF
183
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: Fixed at points A and B,
Inner: None.
Loading
External: None,
Internal: None.
2.67.3 Eigen modes frequencies
Reference solution
The Rayleigh method applied to a thin curved beam is used to determine parameters such as:
transverse bending:
fj = μi
2
2π R2 GIpρA where i = 1,2.
Finite elements modeling
Linear element: beam,
11 nodes,
10 linear elements.
Eigen mode shapes
2.67.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation CM2 Transverse 1 Hz 44.23 44.12 -0.25% CM2 Transverse 2 Hz 125 120.09 -3.93%
Test 01-0068SDLLB_MEF
184
2.68 Test No. 01-0068SDLLB_FEM: Vibration mode of a thin piping elbow in space (case 2) 2.68.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 14/89;
Analysis type: modal analysis (in space);
Element type: linear.
2.68.2 Overview A thin piping elbow with a radius of 1m is extended with two straight elements of length "L" and is loaded with its self weight only.
Vibration mode of a thin piping elbow Scale = 1/11 01-0068SDLLB_FEM
Units I. S.
Geometry
Average radius of curvature: OA = R = 1 m,
L = 0.6 m,
Straight circular hollow section:
Outer diameter: de = 0.020 m,
Inner diameter: di = 0.016 m,
Section: A = 1.131 x 10-4 m2,
Flexure moment of inertia relative to the y-axis: Iy = 4.637 x 10-9 m4,
Flexure moment of inertia relative to z-axis: Iz = 4.637 x 10-9 m4,
Polar inertia: Ip = 9.274 x 10-9 m4.
Points coordinates (in m): o O ( 0 ; 0 ; 0 ) o A ( 0 ; R ; 0 ) o B ( R ; 0 ; 0 ) o C ( -L ; R ; 0 ) o D ( R ; -L ; 0 )
Test 01-0068SDLLB_MEF
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Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: o Fixed at points C and D o In A: translation restraint along y and z, o In B: translation restraint along x and z,
Inner: None.
Loading
External: None,
Internal: None.
2.68.3 Eigen modes frequencies Reference solution
The Rayleigh method applied to a thin curved beam is used to determine parameters such as:
transverse bending:
fj = μi
2
2π R2 GIpρA where i = 1,2.
Finite elements modeling
Linear element: beam,
23 nodes,
22 linear elements.
Eigen mode shapes
2.68.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation CM2 Transverse 1 Hz 33.4 33.19 -0.63% CM2 Transverse 2 Hz 100 94.62 -5.38%
Test 01-0069SDLLB_MEF
186
2.69 Test No. 01-0069SDLLB_FEM: Vibration mode of a thin piping elbow in space (case 3) 2.69.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 14/89;
Analysis type: modal analysis (space problem);
Element type: linear.
2.69.2 Overview A thin piping elbow with a radius of 1m is extended by two straight elements of length "L" and is loaded with its self weight only.
Vibration mode of a thin piping elbow Scale = 1/12 01-0069SDLLB_FEM
Units
I. S.
Geometry
Average radius of curvature: OA = R = 1 m,
L = 2 m,
Straight circular hollow section:
Outer diameter: de = 0.020 m,
Inner diameter: di = 0.016 m,
Section: A = 1.131 x 10-4 m2,
Flexure moment of inertia relative to the y-axis: Iy = 4.637 x 10-9 m4,
Flexure moment of inertia relative to z-axis: Iz = 4.637 x 10-9 m4,
Polar inertia: Ip = 9.274 x 10-9 m4.
Points coordinates (in m): o O ( 0 ; 0 ; 0 ) o A ( 0 ; R ; 0 ) o B ( R ; 0 ; 0 ) o C ( -L ; R ; 0 ) o D ( R ; -L ; 0 )
Test 01-0069SDLLB_MEF
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Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Density: ρ = 7800 kg/m3.
Boundary conditions
Outer: o Fixed at points C and D o At A: translation restraint along y and z, o At B: translation restraint along x and z,
Inner: None.
Loading
External: None,
Internal: None.
2.69.3 Eigen modes frequencies Reference solution
The Rayleigh method applied to a thin curved beam is used to determine parameters such as:
transverse bending:
fj = μi
2
2π R2 GIpρA where i = 1,2 with i = 1,2:
Finite elements modeling
Linear element: beam,
41 nodes,
40 linear elements.
Eigen mode shapes
2.69.4 Results sheet Results comparison: eigen modes frequencies
Solver Eigen mode type Units Reference AD 2010 Deviation CM2 Transverse 1 Hz 17.900 17.65 -1.40% CM2 Transverse 2 Hz 24.800 24.43 -1.49%
Test 01-0077SSLPB_MEF
188
2.70 Test No. 01-0077SSLPB_FEM: Reactions on supports and bending moments on a 2D portal frame (Rafters)
2.70.1 Description sheet
Reference: Design and calculation of metal structures.
Analysis type: static linear;
Element type: linear.
2.70.2 Overview Moments and actions on supports calculation on a 2D portal frame.
The purpose of this test is to verify the results of Advance Design for the M. R. study of a 2D portal frame.
Portal frame geometry
Suppose there is a symmetric portal frame with a range of 20 meters, with columns (of the same inertia as the rafters), that are hinged on 7.5 meters high base plates, and the ridge is at 10 meters altitude. The column and rafters sections are identical.
Portal frame solicitation
The portal frame is successively subjected to:
A linear load of q=100 daN/ml on the rafters, perpendicular to them.
Test 01-0077SSLPB_MEF
189
2.70.3 Moments and actions on supports M.R. calculation on a 2D portal frame.
RDM results, for the linear load perpendicular on the rafters, are:
2qLVV EA == ( ) ( ) H
fh3f3k²hf5h8
32²qLHH EA =
++++
==
HhMM DB −== ( )fhH8
²qLMC +−=
2.70.4 Results sheet
Comparison between theoretical results and the results obtained by Advance Design for a linear load perpendicular on the chords
Solver Forces Units Reference AD 2010 Deviation CM2 Vertical reaction VA = VE DaN -1000 -1000 0.00%
Horizontal reaction HA = HE DaN -332.9 -332.67 -0.07% MB = MD DaN.m -2496.8 -2494.99 -0.07% MC DaN.m -1671 -1673.35 0.14%
Test 01-0078SSLPB_MEF
190
2.71 Test No. 01-0078SSLPB_FEM: Reactions on supports and bending moments on a 2D portal frame (Columns)
2.71.1 Description sheet
Reference: Design and calculation of metal structures.
Analysis type: static linear;
Element type: linear.
2.71.2 Overview Moments and actions on supports calculation on a 2D portal frame.
The purpose of this test is to verify the results of Advance Design for the M. R. study of a 2D portal frame.
Portal frame geometry
Suppose there is a symmetric portal frame with a range of 20 meters, with columns (of the same inertia as the rafters), that are hinged on 7.5 meters high base plates, and the ridge is at 10 meters altitude. The profiles are identical for the column and the rafters.
Portal frame solicitation
The portal frame is successively subjected to:
A linear load of q=100 daN/ml on the column, perpendicular on it.
Test 01-0078SSLPB_MEF
191
2.71.3 Moments and reactions on supports M.R. calculation on a 2D portal frame. RDM results, for the linear load perpendicular on the column, are:
L2²qhVV EA −=−=
( )( ) ( )fh3f3k²h
fh26kh516
²qhHE +++++
= qhHH EA −=
hHqhM EB −=2
² ( )fhH
4²qhM EC +−= hHM ED −=
2.71.4 Results sheet
Comparison between theoretical results and the results obtained by Advance Design for a linear load perpendicular on the column
Solver Forces Units Reference AD 2010 Deviation CM2 Vertical reaction VA = -VE DaN -140.6 -140.63 0.02%
Horizontal reaction HA DaN 579.1 579.17 0.01% Horizontal reaction HE DaN 170.9 170.83 -0.04% MB DaN.m 1530.8 1531.26 0.03% MC DaN.m -302.7 -302.06 -0.21% MD DaN.m -1281.7 -1281.23 -0.04%
Test 01-0084SSLLB_MEF
192
2.72 Test No. 01-0084SSLLB_FEM: Short beam on two hinged supports
2.72.1 Description sheet Reference: Structure Calculation Software Validation Guide, test SSLL 02/89
Analysis type: static linear (plane problem);
Element type: linear.
2.72.2 Overview Short beam on two hinged supports
Units
I. S.
Geometry
Length: L = 1.44 m,
Area: A = 31 x 10-4 m²
Inertia: I = 2810 x 10-8 m4
Shearing coefficient: az = 2.42 = A/Ar
Materials properties
E = 2 x 1011 Pa
ν = 0.3
Boundary conditions
Hinge at end x = 0,
Hinge at end x = 1.44 m.
Loading
Uniformly distributed force of p = -1. X 105 N/m on beam AB.
2.72.3 Reference results Calculation method used to obtain the reference solution
The deflection on the middle of a non-slender beam considering the shear force deformations given by the Timoshenko function:
GA8pl
EIpl
3845v
r
24
+=
where ( )υ+=
12EG and
zr a
AA =
where "Ar" is the reduced area and "az" the shear coefficient calculated on the transverse section.
Uncertainty about the reference: analytical solution:
Test 01-0084SSLLB_MEF
193
Reference values
Point Magnitudes and units Value C V, deflection (m) -1.25926 x 10-3
2.72.4 Results sheet
Results comparison
Solver Point Magnitudes and units Reference value AD 2010 Deviation CM2 C V, deflection (m) -1.25926 -1.25926 0.00%
Test 01-0085SDLLB_MEF
194
2.73 Test No. 01-0085SDLLB_FEM: Slender beam of variable rectangular section with fixed-free ends (β=5)
2.73.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 09/89;
Analysis type: modal analysis (plane problem);
Element type: linear.
2.73.2 Overview Slender beam with variable rectangular section (fixed-free)
Units
I. S.
Geometry
Length: L = 1 m,
Straight initial section: o h0 = 0.04 m o b0 = 0.05 m o A0 = 2 x 10-3 m²
Straight final section o h1 = 0.01 m o b1 = 0.01 m o A1 = 10-4 m²
Materials properties
E = 2 x 1011 Pa
ρ = 7800 kg/m3
Boundary conditions
Outer: o Fixed at end x = 0, o Free at end x = 1
Inner: None.
Loading
External: None,
Internal: None.
Test 01-0085SDLLB_MEF
195
2.73.3 Reference results Calculation method used to obtain the reference solution
Precise calculation by numerical integration of the differential equation of beams bending (Euler-Bernoulli theories):
²t²A
²x²EIz
x2
2
δνδ
ρ−=⎟⎠
⎞⎜⎝
⎛δ
νδδδ
where Iz and A vary with the abscissa.
The result is:
( ) ρβαλπ
= 12E
²l1h,i
21fi with
⎪⎪⎩
⎪⎪⎨
⎧
=ο
=β
=ο
=α
51b
b
41h
h
λ1 λ2 λ3 λ4 λ5
β = 5 24.308 75.56 167.21 301.9 480.4
Uncertainty about the reference: analytical solution:
Reference values
Eigen mode type Frequency (Hz) 1 56.55 2 175.79 3 389.01 4 702.36
Flexion
5 1117.63
MODE 1 Scale = 1/4
Test 01-0085SDLLB_MEF
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MODE 2 Scale = 1/4
MODE 3 Scale = 1/4
Test 01-0085SDLLB_MEF
197
MODE 4 Scale = 1/4
MODE 5 Scale = 1/4
2.73.4 Results sheet
Results comparison
Eigen mode type Frequency Theoretical (Hz)
AD 2010 (Hz)
Deviation
1 56.55 58.49 3.43% 2 175.79 177.67 1.07% 3 389.01 388.85 -0.04% 4 702.36 697.38 -0.71%
Bending
5 1117.63 1106.31 -1.01%
Test 01-0086SDLLB_MEF
198
2.74 Test No. 01-0086SDLLB_FEM: Slender beam of variable rectangular section (fixed-fixed) 2.74.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 10/89;
Analysis type: modal analysis (plane problem);
Element type: linear.
2.74.2 Overview Slender beam with variable rectangular section (fixed-fixed)
Units I. S.
Geometry
Length: L = 0.6 m,
Constant thickness: h = 0.01 m
Initial section: o b0 = 0.03 m o A0 = 3 x 10-4 m²
Section variation: o with (α = 1) o b = b0e-2αx o A = A0e-2αx
Materials properties
E = 2 x 1011 Pa
ν = 0.3
ρ = 7800 kg/m3
Boundary conditions
Outer: o Fixed at end x = 0, o Fixed at end x = 0.6 m.
Inner: None.
Loading
External: None,
Internal: None.
2.74.3 Reference results
Calculation method used to obtain the reference solution
ωi pulsation is given by the roots of the equation:
( ) ( ) ( ) ( ) 0rlsinslshrs2
²r²sslchrlcos1 =−
+−
Test 01-0086SDLLB_MEF
199
with
( ) 0²²s;²r;EIA 2
isi2
i2i
zo
2i04
i >α−λ⎯→⎯α−λ=λ+α=ωρ
=λ
Therefore, the ν translation components of φi(x) mode, are:
( ) ( ) ( ) ⎥⎦
⎤⎢⎣
⎡−
−−
+−=Φ α ))sx(rsh)rxsin(s()rlsin(s)sl(rsh
)sl(ch)rlcos(sxchrxcosex xi
Uncertainty about the reference: analytical solution:
Reference values
Eigen mode φi(x)* Eigen mode order
Frequency (Hz) x = 0 0.1 0.2 0.3 0.4 0.5 0.6
1 143.303 0 0.237 0.703 1 0.859 0.354 0 2 396.821 0 -0.504 -0.818 0 0.943 0.752 0 3 779.425 0 0.670 0.210 -0.831 0.257 1 0 4 1289.577 0 -0.670 0.486 0 -0.594 1 0
* φi(x) eigen modes* standardized to 1 at the point of maximum amplitude.
Eigen modes
2.74.4 Results sheet
Results comparison: eigen modes frequencies
Solver Eigen mode type: Flexion Units Reference AD 2010 Deviation CM2 1 Hz 143.303 145.88 1.80% CM2 2 Hz 396.821 400.26 0.87% CM2 3 Hz 779.425 783.15 0.48% CM2 4 Hz 1289.577 1293.42 0.30%
Test 01-0089SSLLB_MEF
200
2.75 Test No. 01-0089SSLLB_FEM: Plane portal frame with hinged supports
2.75.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SSLL 14/89;
Analysis type: static linear;
Element type: linear.
2.75.2 Overview Calculation of support reactions of a 2D portal frame.
Units
I. S.
Geometry
Length: L = 20 m,
I1 = 5.0 x 10-4 m4
a = 4 m
h = 8 m
b = 10.77 m
I2 = 2.5 x 10-4 m4
Materials properties
Isotropic linear elastic material.
E = 2.1 x 1011 Pa
Boundary conditions
Hinged base plates A and B (uA = vA = 0 ; uB = vB = 0).
Test 01-0089SSLLB_MEF
201
Loading
p = -3 000 N/m
F1 = -20 000 N
F2 = -10 000 N
M = -100 000 Nm
2.75.3 Calculation method used to obtain the reference solution
K = (I2/b)(h/I1)
p = a/h
m = 1 + p
B = 2(K + 1) + m
C = 1 + 2m
N = B + mC
VA = 3pl/8 + F1/2 – M/l + F2h/l
HA = pl²(3 + 5m)/(32Nh) + (F1l/(4h))(C/N) + F2(1-(B + C)/(2N)) + (3M/h)((1 + m)/(2N))
2.75.4 Reference values Point Magnitudes and units Value
A V, vertical reaction (N) 31 500.0 A H, horizontal reaction (N) 20 239.4 C vc (m) -0.03072
2.75.5 Results sheet
Results comparison: force
Solver Point Magnitude Units Reference AD 2010 Deviation CM2 A Vertical reaction V N -31500 -31500 0.00%
A Horizontal reaction H N -20239.4 -20239.4 0.00%
Results comparison: displacement
Solver Point Magnitude Units Reference AD 2010 Deviation CM2 C vc displacement m -0.03072 -0.03072 0.00%
Test 01-0090HFLSB_MEF
202
2.76 Test No. 01-0090HFLSB_FEM: Simply supported beam in Eulerian buckling with a thermal load
2.76.1 Description sheet
Reference: Euler theory;
Analysis type: Eulerian buckling;
Element type: planar.
2.76.2 Overview The 300 cm long beam is a I shaped profile of 20.04 cm height, 0.96 cm thick web and 20.04 cm wide flanges of 1.46 cm thick.
Units
I. S.
Geometry
L = 300 cm h = 20.04 cm b= 20.04 cm tw = 0.96 cm tf = 1.46 cm
Cross Section Sx cm² Sy cm² Sz cm² Ix cm4 Iy cm4 Iz cm4
PRS 74.952 (20.04-0.46) x 0.96=18.58
2x20.04x1.46=58.52
46.54 1959.63 5462.06
Test 01-0090HFLSB_MEF
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Materials properties
Longitudinal elastic modulus: E = 2285938 daN/cm2,
Poisson's ratio: ν = 0.3,
Transverse elastic modulus: G = 879207 daN/cm2,
Coefficient of thermal expansion: α = 0.00001. Boundary conditions
Outer: Webs simply supported at Z = 0 and Z = 300 cm,
Inner: None. Loading
External: None,
Internal: Thermal load ΔT = 100°C on the center corresponding to a normal compression force of: ( ) daN 4077410000001.096.046.104.202285938TESN =×××−×=Δα=
2.76.3 Displacement of the model in the linear elastic range
Reference solution
The reference critical load established by Euler is:
daN 491243300
632.19592285938L
EIN 2
2
2
2
k =××π
=π
=
To take into account the effect of the shear force, we correct this load by:
93.1140774
486597daN 486597N9905.0
GSN
1
NN k
z
k
kk ==λ⇒=×=
+=′
Finite elements modeling
Planar element: Thick shell, imposed mesh,
2025 nodes,
1920 elements.
Thermal efforts
Test 01-0090HFLSB_MEF
204
Deformed shape of mode 1
2.76.4 Results sheet
Results comparison
Solver CASE Magnitude Units Reference AD 2010 Deviation CM2 101 Total normal effort* daN -40774 -41051.95 0.68%
102 Critical coefficient - 11.93 11.873 -0.48% * To obtain this force, select the support from one of the column extremities and display the actions on supports on this selection. This value is displayed in the document along with the values on the selected support. The value is represented by the Fz sum of loads on linear supports per load case - global coordinate system (if the z-axis of the global coordinate system corresponds to the x-axis in the local coordinate system of the bar).
Test 01-0091HFLLB_MEF
205
2.77 Test No. 01-0091HFLLB_FEM: Double fixed beam in Eulerian buckling with a thermal load
2.77.1 Description sheet
Reference: Euler theory; Analysis type: Eulerian buckling; Element type: linear.
2.77.2 Overview
Units
I. S.
Geometry
L = 10 m
Cross Section Sx m² Sy m² Sz m² Ix m4 Iy m4 Iz m4 Vx m3 V1y m3 V1z m3 V2y m3 V2z m3
IPE200 0.002850 0.001400 0.001799 0.0000000646 0.0000014200 0.0000194300 0.00000000 0.00002850 0.00019400 0.00002850 0.00019400
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 N/m2, Poisson's ratio: ν = 0.3. Coefficient of thermal expansion: α = 0.00001
Boundary conditions
Outer: Fixed at end x = 0,
Inner: None.
Test 01-0091HFLLB_MEF
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Loading
External: Punctual load FZ = 1 N at = L/2 (load that initializes the deformed shape),
Internal: ΔT = 5°C corresponding to a compression force of:
kN 925.29500001.000285.011E1.2TESN =×××=Δα=
2.77.3 Displacement of the model in the linear elastic range
Reference solution
The reference critical load established by Euler is:
93.3724.117925.29k 724.117
2
P 2
2
critical ==⇒=
⎟⎠⎞
⎜⎝⎛
= λπ NLEI
Observation: in this case, the thermal load has no effect over the critical coefficient
Finite elements modeling
Linear element: beam, imposed mesh,
1 1 nodes,
10 elements.
Deformed shape of mode 1
2.77.4 Results sheet
Results comparison: normal force
Solver CASE Node Magnitude Units Reference AD 2010 Deviation CM2 101 6 Normal Force kN -29.925 -29.90 -0.08% CM2 102 6 Normal Force kN -117.724 -118.08 0.30%
Test 01-0092HFLLB_MEF
207
2.78 Test No. 01-0092HFLLB_FEM: Cantilever beam in Eulerian buckling with thermal load
2.78.1 Description sheet
Reference: Euler theory; Analysis type: Eulerian buckling; Element type: linear.
2.78.2 Overview
Units
I. S.
Geometry
L = 10 m S=0.01 m2 I = 0.0002 m4
Materials properties
Longitudinal elastic modulus: E = 2.0 x 1010 N/m2, Poisson's ratio: ν = 0.1. Coefficient of thermal expansion: α = 0.00001
Boundary conditions
Outer: Fixed at end x = 0, Inner: None.
Loading
External: Punctual load P = -100000 N at x = L, Internal: T = -50°C (Contraction equivalent to the compression force)
( 5000001.0T0005.001.010.2
100000ESN
100 −×=Δα==×
−==ε )
2.78.3 Displacement of the model in the linear elastic range
Reference solution
The reference critical load established by Euler is:
98696.010000098696 98696
4P 2
2
critical ==⇒== λπ NLEI
Observation: in this case, the thermal load has no effect over the critical coefficient
Test 01-0092HFLLB_MEF
208
Finite elements modeling
Linear element: beam, imposed mesh,
5 nodes,
4 elements.
Deformed shape
2.78.4 Results sheet
Results comparison: normal force
Solver CASE Node Magnitude Units Reference AD 2010 Deviation CM2 101 5 Vertical displacement v5 cm -1.0 -1.0 0.00%
101 A Normal Force N -100000 -100000 0.00% 102 A Normal Force N -98696 -98696 0.00%
Test 01-0094SSLLB_MEF
209
2.79 Test No. 01-0094SSLLB_FEM: 3D bar structure with elastic support 2.79.1 Description sheet
Reference: Internal GRAITEC; Analysis type: static linear; Element type: linear.
2.79.2 Overview Suppose we have the following system of bars:
Units
I. S.
Geometry
For all bars:
H = 3 m B = 3 m S = 0.02 m2
Element Node i Node j 1 (bar) 1 5 2 (bar) 2 5 3 (bar) 3 5 4 (bar) 4 5
5 (spring) 5 6
Materials properties
Isotropic linear elastic materials Longitudinal elastic modulus: E = 2.1 E8 N/m2,
Boundary conditions
Outer: At node 5: K = 50000 kN/m ; Inner: None.
Test 01-0094SSLLB_MEF
210
Loading
External: Vertical load at node: P = -100 kN, Internal: None.
2.79.3 Results
System solution
2
22 BHL += . Also, U1 = V1 = U5 = U6 = V6 = 0
Stiffness matrix of bar 1
( ) ( )
1L2x= where
.121.1
21)(.).1()(
−
++−=⇔+−=
ξ
ξξξ jiji uuuuLxu
Lxxu
in the local coordinate system:
[ ] [ ] [ ][ ] [ ] [ ]
)()()()(
0000010100000101
)()(
1111
41
41
41
41
2=
221
21
2121
1
1
1
101
j
j
i
i
j
i
L Te
T
v
vuvu
LES
uu
LESd
LES
dL
ESdxBBESdVBHBke
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−
=⎥⎦
⎤⎢⎣
⎡−
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
⎭⎬⎫
⎩⎨⎧−
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧−===
∫
∫∫∫
−
−
ξ
ξ
where [ ]
)v()u()v()u(
0000010100000101
LESk
5
5
1
1
1
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−
=
Test 01-0094SSLLB_MEF
211
The elementary matrix [ ]ek expressed in the global coordinate system XY is the following: (θ angle allowing the transition from the global base to the local base):
[ ] [ ] [ ][ ] [ ]
[ ]
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
θθθθ−θθ−θθθθθ−θ−
θ−θθ−θθθθθ−θ−θθθ
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
θθ−θθ
θθ−θθ
==−
22
22
22
22
e
eeeT
ee
sinsincossinsincossincoscossincoscos
sinsincossinsincossincoscossincoscos
LESK
cossin00sincos00
00cossin00sincos
Ravec RkRK
Knowing that LHsin and
2cos == θθ
LB
then:
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
=θθ
=θ
=θ
2
2
22
2
22
L2HBcossin
LHsin
L2Bcos
[ ]
)()()()(
22
2222
22
2222
:)DH(arctan =5,1 nodes 1element for
5
5
1
1
22
22
22
22
31
VUVU
HHBHHB
HBBHBB
HHBHHB
HBBHBB
LESK
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−−
−−
−−
=→ θ
Stiffness matrix of spring support 5
[ ]
)()()()(
0000010100000101
)()(
1111
:system coordinate local in the
4KKsay We
5
j
j
i
i
j
i
vuvu
Kuu
Kk
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−
′=⎥⎦
⎤⎢⎣
⎡−
−′=
=′
[ ]
)()()()(
101000001010
0000
':90=6,5 nodes 5element for
6
6
5
5
5
VUVU
KK
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=°→ θ
Test 01-0094SSLLB_MEF
212
System [ ]{ } { }FQK =
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
−=
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
′′−
′−′+−−
−−
−−
−−
6Y
6X
5X
1Y
1X
6
6
5
5
1
1
233
233
3
2
33
2
3
233
233
3
2
33
2
3
RR
PRRR
VUVUVU
K0K000000000
K0KHLES
2HB
LESH
LES
2HB
LES
002
HBLES
2B
LES
2HB
LES
2B
LES
00HLES
2HB
LESH
LES
2HB
LES
002
HBLES
2B
LES
2HB
LES
2B
LES
If U1 = V1 = U5 = U6 = V6 = 0, then:
m 001885.0
4KH
LES
4P
KHLES
4P
V2
32
3
5 −=+
−=
′+
−=
And N 23563V
4KRN 1436VH
LESR
0RN 1015V2
HBLESRN 1015V
2HB
LESR
56Y52
31Y
6X535X531X
=−==−=
=−===−=
Note:
The values on supports specified by AD 2010 correspond to the actions,
RY6 calculated value must be multiplied by 4 in relation to the double symmetry,
x1 value is similar to the one found by AD 2010 by dividing this by 2
Effort in bar 1:
⎭⎬⎫
⎩⎨⎧
−=
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡−
−
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
17591759
1111
and
200
200
002
002
5
1
5
1
5
5
1
1
5
5
1
1
NN
uu
LES
VUVU
LB
LH
LH
LB
LB
LH
LH
LB
vuvu
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡−
−
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
5
1
5
1
5
5
1
1
5
5
1
1
1111
and
cossin00sincos00
00cossin00sincos
NN
uu
LES
VUVU
vuvu
θθθθ
θθθθ
Test 01-0094SSLLB_MEF
213
Reference values
Point Magnitude Units Value 5 V2 m -1.885 10-3
All bars Normal force N -1759 Supports 1, 3, 4 and 5 Fz action N -1436 Supports 1, 3, 4 and 5 Action Fx=Fy N 7182/1015 ±=
Support 6 Fz action N 23563 x 4=94253
Finite elements modeling
Linear element: beam, automatic mesh,
5 nodes,
4 linear elements.
Test 01-0094SSLLB_MEF
214
Deformed shape
Normal forces diagram
2.79.4 Results sheet
Results comparison
Solver Node Magnitude Units Reference AD 2010 Deviation 5 V2 m -1.885 10-3 -1.885 10-3 0.00%
All bars Normal force N -1759 -1759 0.00% Supports 1, 3, 4 and 5 Fz reaction N -1436 -1436,55 -3.83% Supports 1, 3, 4 and 5 Action Fx=Fy N 7182/1015 ±= 718.27 0.04%
CM2
Support 6 Fz reaction N 23563 x 4=94253 94253.81 0.001%
Test 01-0095SDLLB_MEF
215
2.80 Test No. 01-0095SDLLB_FEM: Fixed/free slender beam with centered mass
2.80.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 15/89;
Analysis type: modal analysis;
Element type: linear.
Tested functions: Eigen mode frequencies, straight slender beam, combined bending-torsion, plane bending, transverse bending, punctual mass.
2.80.2 Test data
Units
I. S.
Geometry
Outer diameter de= 0.35 m,
Inner diameter: di= 0.32 m,
Beam length: l = 10 m,
Area: A= 1.57865 x 10-2 m2
Polar inertia: IP= 4.43798 x 10-4m4
Inertia: Iy= Iz = 2.21899 x 10-4m4
Punctual mass: mc= 1000 kg
Beam self-weight: M
Test 01-0095SDLLB_MEF
216
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 Pa, Density: ρ = 7800 kg/m3
Poisson's ratio: ν=0.3 (this coefficient was not specified in the AFNOR test , the value 0.3 seems to be the more appropriate to obtain the correct frequency value of mode No. 8 with NE/NASTRAN)
Boundary conditions
Outer: Fixed at point A, x = 0, Inner: none
Loading
None for the modal analysis
2.80.3 Reference results
Reference frequency
For the first mode, the Rayleigh method gives the approximation formula
)M24.0m(IEI3
x2/1fc
3z
1 +Π=
Mode Shape Units Reference 1 Flexion Hz 1.65 2 Flexion Hz 1.65 3 Flexion Hz 16.07 4 Flexion Hz 16.07 5 Flexion Hz 50.02 6 Flexion Hz 50.02 7 Traction Hz 76.47 8 Torsion Hz 80.47 9 Flexion Hz 103.2 10 Flexion Hz 103.2
Comment: The mass matrix associated with the beam torsion on two nodes, is expressed as:
⎥⎦
⎤⎢⎣
⎡×
××ρ12/12/11
3Il P
And to the extent that Advance Design 2010 uses a condensed mass matrix, the value of the torsion mass inertia
introduced in the model is set to: 3
Il p××ρ
Uncertainty about the reference frequencies
Analytical solution mode 1 Other modes: ± 1%
Finite elements modeling
Linear element AB: Beam Beam meshing: 20 elements.
Test 01-0095SDLLB_MEF
217
Modal deformations
Test 01-0095SDLLB_MEF
218
Observation: the deformed shape of mode No. 8 that does not really correspond to a torsion deformation, is actually the display result of the translations and not of the rotations. This is confirmed by the rotation values of the corresponding mode.
Eigen modes vector 8
Node DX DY DZ RX RY RZ 1 -3.336e-033 6.479e-031 -6.316e-031 1.055e-022 5.770e-028 5.980e-028 2 -5.030e-013 1.575e-008 -1.520e-008 1.472e-002 6.022e-008 6.243e-008 3 -1.005e-012 6.185e-008 -5.966e-008 2.944e-002 1.171e-007 1.214e-007 4 -1.505e-012 1.365e-007 -1.317e-007 4.416e-002 1.705e-007 1.769e-007 5 -2.002e-012 2.381e-007 -2.296e-007 5.887e-002 2.206e-007 2.289e-007 6 -2.495e-012 3.648e-007 -3.517e-007 7.359e-002 2.673e-007 2.774e-007 7 -2.983e-012 5.149e-007 -4.963e-007 8.831e-002 3.106e-007 3.225e-007 8 -3.464e-012 6.867e-007 -6.618e-007 1.030e-001 3.506e-007 3.641e-007 9 -3.939e-012 8.785e-007 -8.464e-007 1.177e-001 3.873e-007 4.023e-007 10 -4.406e-012 1.088e-006 -1.049e-006 1.325e-001 4.207e-007 4.371e-007 11 -4.863e-012 1.315e-006 -1.267e-006 1.472e-001 4.508e-007 4.684e-007 12 -5.310e-012 1.556e-006 -1.499e-006 1.619e-001 4.777e-007 4.964e-007 13 -5.746e-012 1.811e-006 -1.744e-006 1.766e-001 5.015e-007 5.210e-007 14 -6.169e-012 2.077e-006 -2.000e-006 1.913e-001 5.221e-007 5.423e-007 15 -6.580e-012 2.353e-006 -2.265e-006 2.061e-001 5.396e-007 5.605e-007 16 -6.976e-012 2.637e-006 -2.539e-006 2.208e-001 5.541e-007 5.755e-007 17 -7.357e-012 2.928e-006 -2.819e-006 2.355e-001 5.658e-007 5.874e-007 18 -7.723e-012 3.224e-006 -3.104e-006 2.502e-001 5.746e-007 5.965e-007 19 -8.072e-012 3.524e-006 -3.393e-006 2.649e-001 5.808e-007 6.028e-007 20 -8.403e-012 3.826e-006 -3.685e-006 2.797e-001 5.844e-007 6.065e-007 21 -8.717e-012 4.130e-006 -3.977e-006 2.944e-001 5.856e-007 6.077e-007
With NE/NASTRAN, the results associated with mode No. 8, are:
Test 01-0095SDLLB_MEF
219
2.80.4 Results sheet
Results comparison
Solver Mode Shape Units Reference AD 2010 Deviation (%)
1 Flexion Hz 1.65 1.65 0.00% 2 Flexion Hz 1.65 1.65 0.00% 3 Flexion Hz 16.07 16.06 -0.06% 4 Flexion Hz 16.07 16.06 -0.06%
CM2 5 Flexion Hz 50.02 50.00 -0.04% 6 Flexion Hz 50.02 50.00 -0.04% 7 Traction Hz 76.47 76.46 -0.01% 8 Torsion Hz 80.47 9 Flexion Hz 103.20 103.14 -0.06% 10 Flexion Hz 103.20 103.14 -0.06%
Comment: The difference between the reference frequency of torsion mode (mode No. 8) and the one found by AD 2010 may be explained by the fact that AD 2010 is using a lumped mass matrix (see the corresponding description sheet).
Test 01-0096SDLLB_MEF
220
2.81 Test No. 01-0096SDLLB_FEM: Fixed/free slender beam with eccentric mass or inertia
2.81.1 Description sheet
Reference: Structure Calculation Software Validation Guide, test SDLL 15/89; Analysis type: modal analysis; Element type: linear. Tested functions: Eigen mode frequencies, straight slender beam, combined bending-torsion, plane bending,
transverse bending, punctual mass..
2.81.2 Problem data
Units I. S.
Geometry
Outer diameter: de= 0.35 m, Inner diameter: di = 0.32 m, Beam length: l = 10 m, Distance BC: lBC = 1 m Area: A =1.57865 x 10-2 m2
Inertia: Iy = Iz = 2.21899 x 10-4m4 Polar inertia: Ip = 4.43798 x 10-4m4 Punctual mass: mc = 1000 kg
Materials properties
Longitudinal elasticity modulus of AB element: E = 2.1 x 1011 Pa, Density of the linear element AB: ρ = 7800 kg/m3 Poisson's ratio ν=0.3(this coefficient was not specified in the AFNOR test , the value 0.3 seems to be the more
appropriate to obtain the correct frequency value of modes No. 4 and 5 with NE/NASTRAN: Elastic modulus of BC element: E = 1021 Pa Density of the linear element BC: ρ = 0 kg/m3
Test 01-0096SDLLB_MEF
221
Boundary conditions
Fixed at point A, x = 0,
Loading None for the modal analysis
2.81.3 Reference frequencies
Reference solutions
The different eigen frequencies are determined using a finite elements model of Euler beam (slender beam).
fz + t0 = flexion x,z + torsion
fy + tr = flexion x,y + traction
Mode Units Reference 1 (fz + t0) Hz 1.636 2 (fy + tr) Hz 1.642 3 (fy + tr) Hz 13.460 4 (fz + t0) Hz 13.590 5 (fz + t0) Hz 28.900 6 (fy + tr) Hz 31.960 7 (fz + t0) Hz 61.610 1 (fz + t0) Hz 63.930
Uncertainty about the reference solutions The uncertainty about the reference solutions: ± 1%
Finite elements modeling
Linear element AB: Beam
Imposed mesh: 50 elements.
Linear element BC: Beam
Without meshing
Modal deformations
Test 01-0096SDLLB_MEF
222
2.81.4 Results sheet
Results comparison
Solver Mode Units Reference AD 2010 Deviation (%)
1 (fz + t0) Hz 1.636 1.636 0.00% 2 (fy + tr) Hz 1.642 1.649 0.43% 3 (fy + tr) Hz 13.46 13.455 -0.04% 4 (fz + t0) Hz 13.59 13.647 0.42%
CM2 5 (fz + t0) Hz 28.90 29.72 2.84% 6 (fy + tr) Hz 31.96 31.96 0.00% 7 (fz + t0) Hz 61.61 63.091 2.40% 8 (fy + tr) Hz 63.93 63.93 0.00%
Note:
fz + t0 = flexion x,z + torsion
fy + tr = flexion x,y + traction
Observation: because the mass matrix of AD 2010 is condensed and not consistent, the torsion modes obtained
are not taking into account the self rotation mass inertia of the beam.
Test 01-0097SDLLB_MEF
223
2.82 Test No. 01-0097SDLLB_FEM: Double cross with hinged ends 2.82.1 Description sheet
Reference: NAFEMS, FV2 test Analysis type: modal analysis; Tested functions: Eigen frequencies, Crossed beams, In plane bending.
2.82.2 Problem data
Units I. S.
Geometry
Full square section:
Arm length: L = 5 m Dimensions: a x b = 0.125 x 0.125 Area: A = 1.563 10-2 m2 Inertia: IP = 3.433 x 10-5m4
Iy = Iz = 2.035 x 10-5m4
Materials properties
Longitudinal elastic modulus: E = 2 x 1011 Pa, Density: ρ = 8000 kg/m3
Boundary conditions
Outer: A, B, C, D, E, F, G, H points restraint along x and y; Inner: None.
Loading None for the modal analysis
Test 01-0097SDLLB_MEF
224
2.82.3 Reference frequencies Mode Units Reference
1 Hz 11.336 2,3 Hz 17.709
4 to 8 Hz 17.709 9 Hz 45.345
10,11 Hz 57.390 12 to 16 Hz 57.390
Finite elements modeling
Linear elements type: Beam
Imposed mesh: 4 Elements / Arms
Modal deformations
Test 01-0097SDLLB_MEF
225
2.82.4 Results sheet
Results comparison
Solver Mode Units Reference AD 2010 Deviation (%)
1 Hz 11.336 11.333 -0.03% 2, 3 Hz 17.709 17.662 -0.27%
CM2 4 to 8 Hz 17.709 17.691 -0.10% 9 Hz 45.345 45.016 -0.73% 10, 11 Hz 57.390 56.059 -2.32% 12 to 16 Hz 57.390 56.344 -1.82%
Test 01-0098SDLLB_MEF
226
2.83 Test No. 01-0098SDLLB_FEM: Simple supported beam in free vibration 2.83.1 Description sheet
Reference: NAFEMS, FV5 Analysis type: modal analysis; Tested functions: Shear force, eigen frequencies.
2.83.2 Problem data
Units
I. S.
Geometry
Full square section:
Dimensions: a x b = 2m x 2 m Area: A = 4 m2 Inertia: IP = 2.25 m4
Iy = Iz = 1.333 m4
Materials properties
Longitudinal elastic modulus: E = 2 x 1011 Pa, Poisson's ratio: ν = 0.3. Density: ρ = 8000 kg/m3
Boundary conditions
Outer: o x = y = z = Rx = 0 at A ; o y = z =0 at B ;
Inner: None.
Test 01-0098SDLLB_MEF
227
Loading
None for the modal analysis
2.83.3 Reference frequencies Mode Shape Units Reference
1 Flexion Hz 42.649 2 Flexion Hz 42.649 3 Torsion Hz 77.542 4 Traction Hz 125.00 5 Flexion Hz 148.31 6 Flexion Hz 148.31 7 Torsion Hz 233.10 8 Flexion Hz 284.55 9 Flexion Hz 284.55
Comment: Due to the condensed (lumped) nature of the mass matrix of AD 2010, the frequencies values of 3 and 7 modes cannot be found by this software. The same modeling done with NE/NASTRAN gave respectively for mode 3 and 7: 77.2 and 224.1 Hz.
Finite elements modeling
Straight elements: linear element
Imposed mesh: 5 meshes
Modal deformations
Test 01-0098SDLLB_MEF
228
2.83.4 Results sheet
Results comparison
Solver Mode Shape Units Reference AD 2010 Deviation (%) 1 Flexion Hz 42.649 43.11 1.08% 2 Flexion Hz 42.649 43.11 1.08% 3 Torsion Hz 77.542 4 Tension Hz 125.00 124.49 -0.41%
CM2 5 Flexion Hz 148.31 149.39 0.73% 6 Flexion Hz 148.31 149.39 0.73% 7 Torsion Hz 233.10 8 Flexion Hz 284.55 269.55 -5.27% 9 Flexion Hz 284.55 269.55 -5.27%
Comment: The torsion modes No. 3 and 7 that are calculated with NASTRAN cannot be calculated with Advance
Design CM2 solver and therefore the mode No. 3 of the Advance Design analysis corresponds to mode No. 4 of the reference. The same problem in the case of No. 7 - Advance Design, that corresponds to mode No. 8 of the reference.
Test 01-0099HSLSB_MEF
229
2.84 Test No. 01-0099HSLSB_FEM: Membrane with hot point
2.84.1 Description sheet
Reference: NAFEMS, Test T1
Analysis type: static, thermo-elastic;
Tested functions: Stresses.
2.84.2 Problem data
Observation: the units system of the initial NAFEMS test, defined in mm, was transposed in m for practical reasons. However, this has no influence on the results values.
Units
I. S.
Test 01-0099HSLSB_MEF
230
Geometry / meshing
A quarter of the structure is modeled by incorporating the terms of symmetries.
Thickness: 1 m
Materials properties
Longitudinal elastic modulus: E = 1 x 1011 Pa,
Poisson's ratio: ν = 0.3,
Elongation coefficient α = 0.00001.
Boundary conditions
Outer: o For all nodes in y = 0, uy =0; o For all nodes in x = 0, ux =0;
Inner:None.
Loading
External: None,
Internal: Hot point, thermal load ΔT = 100°C;
Test 01-0099HSLSB_MEF
231
2.84.3 σyy stress at point A:
Reference solution:
Reference value: σyy = 50 MPa in A
Finite elements modeling
Planar elements: membranes,
28 planar elements,
39 nodes.
2.84.4 Results sheet
Results comparison
Solver Quantity Units Reference AD 2010 Deviation (%) CM2 σyy in A MPa 50 50.87 1.74%
Note: This value (50.87) is obtained with a vertical cross section through point A. The value represents σyy at the
left end of the diagram.
With CM2, it is essential to display the results with the “Smooth results on planar elements” option deactivated.
A – hot point limit
Test 01-0100SSNLB_MEF
232
2.85 Test No. 01-0100SSNLB_FEM: Beam on 3 supports with T/C (k = 0)
2.85.1 Description sheet
Reference: internal GRAITEC test; Analysis type: static non linear; Element type: linear, T/C.
2.85.2 Overview Consider the beam on 3 supports as described below. This beam consists of two elements of the same length and identical characteristics.
Units
I. S.
Geometry
L = 10 m Section: IPE 200, Iz = 0.00001943 m4
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 N/m2, Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: o Support at node 1 restrained along x and y (x = 0), o Support at node 2 restrained along y (x = 10 m), o T/C stiffness ky = 0,
Inner: None.
Loading
External: Vertical punctual load P = -100 N at x = 5 m, Internal: None.
Test 01-0100SSNLB_MEF
233
2.85.3 References solutions ky being null, the non linear model behaves the same way as the structure without support 3.
Displacements
( )( )
( )( )
( )( )
( ) rad 000153.0Lk2EI3EI32
LkEI6PL
m 00153.0Lk2EI316
PL3v
rad 000153.0Lk2EI3EI16
LkEI3PL
rad 000153.0Lk2EI3EI32
LkEI2PL3
3yzz
3yz
2
3
3yz
3
3
3yzz
3yz
2
2
3yzz
3yz
2
1
=+
+−=β
=+
−=
=+
+−=β
−=+
+=β
Mz Moments
( )( )
N.m 2502
MM4
PL)m5x(M
0Lk2EI316
PLk3M
0M
1z2zz
3yz
4y
2z
1z
−=−
+==
=+
=
=
Finite elements modeling
Linear element: S beam, automatic mesh,
3 nodes,
2 linear elements + 1 T/C.
Deformed shape
Test 01-0100SSNLB_MEF
234
Moment diagrams
2.85.4 Results sheet
Results comparison
Solver Nature Positioning Units Reference AD 2010 Deviation
β1 rotation Node 1 rad -0.000153 -0.000153 0.00% β2 rotation Node 2 rad 0.000153 0.000153 0.00% v3 displacement Node 3 m 0.00153 0.00153 0.00%
CM2 β3 rotation Node 3 rad 0.000153 0.000153 0.00% Mz1 moment Node 1 N.m 0 0 - Mz2 moment Node 2 N.m 0 0 - Mz moment middle span 1 N.m -250 -250 0.00%
Test 01-0101SSNLB_MEF
235
2.86 Test No. 01-0101SSNLB_FEM: Beam on 3 supports with T/C (k → ∞)
2.86.1 Description sheet
Reference: internal GRAITEC test; Analysis type: static non linear; Element type: linear, T/C.
2.86.2 Overview Consider the beam on 3 supports described below. This beam consists of two elements of the same length and identical characteristics.
Units
I. S.
Geometry
L = 10 m Section: IPE 200, Iz = 0.00001943 m4
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 N/m2, Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: o Support at node 1 restrained along x and y (x = 0), o Support at node 2 restrained along y (x = 10 m), o T/C stiffness ky → ∞ (1.1030N/m),
Inner: None.
Loading
External: Vertical punctual load P = -100 N at x = 5 m, Internal: None.
Test 01-0101SSNLB_MEF
236
2.86.3 References solutions ky being infinite, the non linear model behaves the same way as a beam on 3 supports.
Displacements
( )( )
( )( )
( )( )
( ) rad 000038.0Lk2EI3EI32
LkEI6PL
0Lk2EI316
PL3v
rad 000077.0Lk2EI3EI16
LkEI3PL
rad 000115.0Lk2EI3EI32
LkEI2PL3
3yzz
3yz
2
3
3yz
3
3
3yzz
3yz
2
2
3yzz
3yz
2
1
−=+
+−=β
=+
−=
=+
+−=β
−=+
+=β
Mz Moments
( )( )
N.m 13.2032
MM4
PL)m5x(M
N.m 75.93Lk2EI316
PLk3M
0M
1z2zz
3yz
4y
2z
1z
−=−
+==
−=+
=
=
Finite elements modeling
Linear element: S beam, automatic mesh,
3 nodes,
2 linear elements + 1 T/C.
Deformed shape
Test 01-0101SSNLB_MEF
237
Moment diagram
2.86.4 Results sheet
Results comparison
Solver Nature Positioning Units Reference AD 2010 Deviation
β1 rotation Node 1 rad -0.000115 -0.000115 0.00% β2 rotation Node 2 rad 0.000077 0.000077 0.00% v3 displacement Node 3 m 0 0 -
CM2 β3 rotation Node 3 rad -0.000038 -0.000038 0.00% Mz1 moment Node 1 N.m 0 0 - Mz2 moment Node 2 N.m -93.75 -93.649 -0.11% Mz moment middle span 1 N.m -203.13 -203.176 0.02%
Test 01-0102SSNLB_MEF
238
2.87 Test No. 01-0102SSNLB_FEM: Beam on 3 supports with T/C (k = -10000 N/m)
2.87.1 Description sheet
Reference: internal GRAITEC test;
Analysis type: static non linear;
Element type: linear, T/C.
2.87.2 Overview Consider the beam on 3 supports, as described below. This beam consists of two elements of the same length and identical characteristics.
Units
I. S.
Geometry
L = 10 m
Section: IPE 200, Iz = 0.00001943 m4
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 N/m2,
Poisson's ratio: ν = 0.3.
Boundary conditions
Outer: o Support at node 1 restrained along x and y (x = 0), o Support at node 2 restrained along y (x = 10 m), o T/C ky Rigidity = -10000 N/m (the – sign corresponds to an upwards restraint),
Inner: None.
Loading
External: Vertical punctual load P = -100 N at x = 5 m,
Internal: None.
Test 01-0102SSNLB_MEF
239
2.87.3 References solutions
Displacements
( )( )
( )( )
( )( )
( ) rad 000034.0Lk2EI3EI32
LkEI6PL
m 00058.0Lk2EI316
PL3v
rad 000106.0Lk2EI3EI16
LkEI3PL
rad 000129.0Lk2EI3EI32
LkEI2PL3
3yzz
3yz
2
3
3yz
3
3
3yzz
3yz
2
2
3yzz
3yz
2
1
=+
+−=β
=+
−=
=+
+−=β
−=+
+=β
Mz Moments
( )( )
N.m 9.2202
MM4
PL)m5x(M
N.m 15.58Lk2EI316
PLk3M
0M
1z2zz
3yz
4y
2z
1z
−=−
+==
−=+
=
=
Finite elements modeling
Linear element: S beam, automatic mesh,
3 nodes,
2 linear elements + 1 T/C.
Deformed shape
Test 01-0102SSNLB_MEF
240
Moment diagram
2.87.4 Results sheet
Results comparison
Solver Nature Positioning Units Reference AD 2010 Deviation
β1 rotation Node 1 rad -0.000129 -0.000129 0.00% β2 rotation Node 2 rad 0.000106 0.000106 0.00% v3 displacement Node 3 m 0.00058 0.00058 0.00%
CM2 β3 rotation Node 3 rad 0.000034 0.000034 0.00% Mz1 moment Node 1 N.m 0 0 - Mz2 moment Node 2 N.m -58.15 -58.117 -0.06% Mz moment middle span 1 N.m -220.9 -220.9 0.00%
Test 01-0103SSLLB_MEF
241
2.88 Test No. 01-0103SSLLB_FEM: Linear system of truss beams
2.88.1 Description sheet
Reference: internal GRAITEC test; Analysis type: static linear; Element type: linear, bar.
2.88.2 Overview Consider the bar system described below. This system contains 4 elements of the same length and S sections (bars
1 to 4) and 2 diagonals of 2L length and 2
S section (bars 5 and 6).
Units
I. S.
Geometry
L = 5 m Section S = 0.005 m2
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 N/m2.
Boundary conditions
Outer: o Support at node 1 restrained along x and y, o Support at node 2 restrained along x and y,
Inner: None.
Loading
External: Horizontal punctual load P = 50000 N at node 3, Internal: None.
Test 01-0103SSLLB_MEF
242
2.88.3 References solutions
Displacements
m 000108.0ES11PL5v
m 000541.0ES11PL25u
m 000129.0ES11PL6v
m 000649.0ES11PL30u
4
4
3
3
==
==
−=−
=
==
N normal forces
N 32141P11
25N N 22727P115N
N 38569P11
26N N 27272P116N
N 22727P115N 0N
4243
1323
1412
−=−===
==−=−=
===
Finite elements modeling
Linear element: bar, without meshing,
4 nodes,
6 linear elements.
Deformed shape
Test 01-0103SSLLB_MEF
243
Normal forces
2.88.4 Results sheet
Results comparison
Solver Nature Positioning Units Reference AD 2010 Deviation
u3 displacement Node 3 m 0.000649 0.000649 0.00% v3 displacement Node 3 m -0.000129 -0.000129 0.00% u4 displacement Node 4 m 0.000541 0.000541 0.00% v4 displacement Node 4 m 0.000108 0.000108 0.00%
CM2 N12 normal force Element 1 N 0 0 - N23 normal force Element 2 N -27272 -27272 0.00% N43 normal force Element 3 N 22727 22727 0.00% N14 normal force Element 4 N 22727 22727 0.00% N13 normal effort Element 5 N 38569 38569 0.00% N42 normal force Element 6 N -32141 -32141 0.00%
Test 01-0104SSNLB_MEF
244
2.89 Test No. 01-0104SSNLB_FEM: Non linear system of truss beams
2.89.1 Description sheet
Reference: internal GRAITEC test; Analysis type: static non linear; Element type: linear, bar, tie.
2.89.2 Overview Consider the bar system described below. This system contains 4 elements of the same length and S sections (bars
1 to 4) and 2 diagonals of 2L length and 2
S section (ties 5 and 6).
Units I. S.
Geometry
L = 5 m Section S = 0.005 m2
Materials properties
Longitudinal elastic modulus: E = 2.1 x 1011 N/m2.
Boundary conditions
Outer: o Support at node 1 restrained along x and y, o Support at node 2 restrained along x and y,
Inner: None.
Loading
External: Horizontal punctual load P = 50000 N at node 3, Internal: None.
Test 01-0104SSNLB_MEF
245
2.89.3 References solutions In non linear analysis without large displacement, the introduction of ties for the diagonal bars removes bar 5 (test No. 0103SSLLB_FEM allows finding an compression force in this bar at the linear calculation).
Displacements
0v
m 000238.0ESPLv
m 001195.0ES11PL5uu
4
3
43
=
−=−=
===
N normal forces
0N 0NN 70711P2N N 50000PN
0N 0N
4243
1323
1412
==
==−=−=
==
Finite elements modeling
Linear element: bar, without meshing, 4 nodes, 6 linear elements.
Deformed shape
Test 01-0104SSNLB_MEF
246
Normal forces
2.89.4 Results sheet
Results comparison
Solver Nature Positioning Units Reference AD 2010 Deviation
u3 displacement Node 3 m 0.001195 0.001190 -0.42% v3 displacement Node 3 m -0.000238 -0.000238 0.00% u4 displacement Node 4 m 0.001195 0.001190 -0.42% v4 displacement Node 4 m 0 0 -
CM2 N12 normal force Element 1 N 0 0 - N23 normal force Element 2 N -50000 -50000 0.00% N43 normal force Element 3 N 0 0 - N14 normal force Element 4 N 0 0 - N13 normal force Element 5 N 70711 70711 0.00% N42 normal force Element 6 N 0 0 -
Test 02-0112SMLLB_P92
247
2.90 Test No. 02-0112SMLLB_P92: Study of a mast subjected to an earthquake
2.90.1 Description sheet
Reference: internal GRAITEC test; Analysis type: modal and spectral analyses; Element type: linear, mass.
2.90.2 Model overview The structure below consists of 2 beams and 2 punctual masses, subject to a lateral earthquake along X.
2.90.3 Material strength model
Units
I. S.
Geometry
Length: L = 35 m,
Outer radius: Rext = 3.00 m
Inner radius: Rint = 2.80 m
Axial section: S= 3.644 m2
Polar inertia: Ip = 30.68 m4
Bending inertias: Ix =15.34 m4 Iy = 15.34 m4
Masses
M1 =203873.6 kg
M2 =101936.8 kg
Materials properties
Longitudinal elastic modulus: E = 1.962 x 1010 N/m2,
Poisson's ratio: ν = 0.1,
Density: ρ = 25 kN/m3
Test 02-0112SMLLB_P92
248
Boundary conditions
Outer: Fixed in X = 0, Y = 0 m,
Loading
External: Seismic excitation on X direction
Finite elements modeling
Linear element: beam, automatic mesh,
2.90.4 Seismic hypothesis in conformity with PS92 regulation
Zone: Nice Sophia Antipolis (Zone II).
Site: S1 (Medium soil, 10m thickness).
Construction type class: B
Behavior coefficient: 3
Material damping: 4% (Reinforced concrete).
2.90.5 Modal analysis
Eigen periods reference solution
Substract the value of structure’s specific horizontal periods by solving the following equation:
( ) 0MKdet 2 =ω−
⎟⎟⎠
⎞⎜⎜⎝
⎛≡
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−≡
2
1
3
M00M
M
25516
L7EI48K
Eigen modes Units Reference 1 Hz 2.085 2 Hz 10.742
Modal vectors
For ω1:
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛χ⇒=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
ωω
−⎟⎟⎠
⎞⎜⎜⎝
⎛−
−055.31
UU
0UU
M00M
25516
L7EI48
2
11
2
1
12
2
211
3
For ω2:
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=⎟⎟⎠
⎞⎜⎜⎝
⎛χ
655.01
UU
2
12
Normalizing relative to the mass
⎟⎟⎠
⎞⎜⎜⎝
⎛
××
χ−
−
3
4
1 10842.210305.9
; ⎟⎟⎠
⎞⎜⎜⎝
⎛
×−×
χ−
−
3
3
2 10316.11001.2
Test 02-0112SMLLB_P92
249
Modal deformations
2.90.6 Spectral study
Design spectrum
Nominal acceleration:
2n1 sm5411.5a 2.085Hzf =⇒=
2n2 sm25.6aHz742.01f =⇒=
Observation: the gap between pulses is greater than 10%, so the modal responses can be regarded as independent.
Test 02-0112SMLLB_P92
250
Reference participation factors
Δχ=γ Mii
séismedudirectionladedirecteurVecteur: ⋅⋅⋅⋅⋅⋅Δ
Eigen modes Reference 1 479.427 2 275.609
Pseudo-acceleration
iiii a χ×γ×ξ×=Γ in (m/s2)
4.0%5
⎟⎟⎠
⎞⎜⎜⎝
⎛η
=ξ : Damping correction factor.
η: Structure damping.
⎟⎟⎠
⎞⎜⎜⎝
⎛=Γ
8.25562.7026
1 ⎟⎟⎠
⎞⎜⎜⎝
⎛=Γ
2.4783-3.7852
1
Reference modal displacement
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=ψ024.814E021.576E
1 ⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=ψ
045.446E-048.318E
2
Equivalent static forces
⎟⎟⎠
⎞⎜⎜⎝
⎛++
=058.415E055.510E
F1 ⎟⎟⎠
⎞⎜⎜⎝
⎛+
+=
052.526E-057.717E
F2
Displacement at the top of the mast
( ) ( )( )221 04E446.502E81.4U −−+−=
Units Reference m 4.814 E-02
Shear force at the top of the mast
( ) ( )( )3
05E526.205E415.8T
22
1+−++
=
3: Being the behavior coefficient of forces
Units Reference N 2.929 E+05
Moment at the base
( ) ( )( ) ( ) ( )( )3
05E717.705E510.55.1705E526.205E415.835M
2222 +++×++++×=
Units Reference N.m 1.578 E+07
Test 02-0112SMLLB_P92
251
2.90.7 Results sheet
Results comparison: Frequencies
Solver Modes Units Reference AD 2010 Deviation CM2 1 Hz 2.085 2.085 0.00% CM2 2 Hz 10.742 10.737 -0.05%
Results comparison: Eigen vectors
Solver Modes Node Reference AD 2010 Deviation CM2 1 9.305 E-04 9.305 E-04 0.00% CM2 1 2 2.842 E-03 2.842 E-03 0.00% CM2 1 2.010 E-03 2.010 E-03 0.00% CM2 2 2 -1.316 E-03 -1.316 E-03 0.00%
Results comparison: Participation factors
Solver Modes Reference AD 2010 Deviation CM2 1 479.43 479.43 0.00% CM2 2 275.61 275.61 0.00%
Results comparison: Displacement at the top of the mast
Solver Units Reference AD 2010 Deviation CM2 cm 4.814 4.812 -0.04%
Results comparison: Forces at the top of the mast
Solver Units Reference AD 2010 Deviation CM2 N 2.929E+05 2.927E+5 -0.07%
Test 02-0158SSLLB_B91
252
2.91 Test No. 02-0158SSLLB_B91: Linear element in combined bending/tension - without compressed reinforcements - Partially tensioned section
2.91.1 Description sheet
Reference: J. Perchat (CHEC) reinforced concrete course
Analysis type: static linear;
Element type: planar.
2.91.2 Overview Beam with 8 isostatic spans subjects to uniform loads and compression normal forces.
Units
Forces: kN
Moment: kN.m
Stresses: MPa
Reinforcement density: cm²
Geometry
Beam dimensions: 0.2 x 0.5 ht
Length: l = 48 m in 8 spans of 6m,
Materials properties
Longitudinal elastic modulus: E = 20000 MPa,
Poisson's ratio: ν = 0.
Boundary conditions
Outer: o Hinged at end x = 0, o Vertical support at the same level with all other supports
Inner: Hinged at each beam end (isostatic)
Loading
External: o Case 1 (DL): uniform linear load g= -5kN/m (on all spans except 8)
Fx = 10 kN at x = 42m: Ng = -10 kN for spans from 6 to 7 Fx = 140 kN at x = 32m: Ng = -150 kN for span 5 Fx = -50 kN at x = 24m: Ng = -100 kN for span 4 Fx = 50 kN at x = 18m: Ng = -50 kN for span 3 Fx = 50 kN at x = 12m: Ng = -100 kN for span 2 Fx = -70 kN at x = 6m: Ng = -30 kN for span 1
o Case 10 (DL): uniform linear load g = -5 kN/m (span 8) Fx = 10 kN at x = 48m: Ng = -10 kN Fx = -10 kN at x = 42m
Test 02-0158SSLLB_B91
253
o Case 2 to 8 (LL): uniform linear load q = -9 kN/m (on spans 1, 3 to 7) uniform linear load q = -15 kN/m (on span 2) Fx = 30 kN at x = 6m (case 2 span 1) Fx = -50 kN at x = 6m (case 3 span 2) Fx = 50 kN at x = 12m (case 3 span 2) Fx = -40 kN at x = 12m (case 4 span 3) Fx = 40 kN at x = 18m (case 4 span 3) Fx = -100 kN at x = 18m (case 5 span 4) Fx = 100 kN at x = 24m (case 5 span 4) Fx = -150 kN at x = 24m (case 6 span 5) Fx = 150 kN at x = 30m (case 6 span 5) Fx = -8 kN at x = 30m (case 7 span 6) Fx = 8 kN at x = 36m (case 7 span 6) Fx = -8 kN at x = 36m (case 8 span 7) Fx = 8 kN at x = 42m (case 8 span 7)
o Case 9 (ACC): uniform linear load a = -25 kN/m (on 8th span) Fx = 8 kN at x = 36m (case 9 span 8) Fx = -8 kN at x = 42m (case 9 span 8) Comb BAELUS: 1.35xDL+1.5xLL with duration of more than 24h (comb 101, 104 to 107) Comb BAEULI: 1.35xDL+1.5xLL with duration between 1h and 24h (comb 102) Comb BAELUC: 1.35xDL + 1.5xLL with duration of less than 1h (comb 103) Comb BAELS: 1xDL + 1*LL (comb 108 to 114) Comb BAELA: 1xDL + 1xACC with duration of less than 1h (comb 115)
Internal: None.
Reinforced concrete calculation hypothesis: All concrete covers are set to 5 cm
BAEL 91 calculation (according to 99 revised version)
Span Concrete Reinforcement Application Concrete Cracking 1 B20 HA fe500 D>24h No Non
prejudicial 2 B35 Adx fe235 1h<D<24h No Non
prejudicial 3 B50 HA fe 400 D<1h Yes Non
prejudicial 4 B25 HA fe500 D>24h Yes Prejudicial 5 B25 HA fe500 D>24h No Very
prejudicial 6 B30 Adx fe235 D>24h Yes Prejudicial 7 B40 HA fe500 D>24h Yes 160 MPa 8 B45 HA fe500 D<1h Yes Non
prejudicial
2.91.3 Reinforcement calculation
Reference solution
Span 1 Span 2 Span 3 Span 4 Span 5 Span 6 Span 7 Span 8 fc28 20 35 50 25 25 30 40 45 ft28 1.8 2.7 3.6 2.1 2.1 2.4 3 3.3 fe 500 235 400 500 500 235 500 500
teta 1 0.9 0.85 1 1 1 1 0.85 gamb 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.15 gams 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1
h 1.6 1 1.6 1.6 1.6 1 1.6 1.6
fbu 11.33 22.04 33.33 14.17 14.17 17.00 22.67 39.13 fed 434.78 204.35 347.83 434.78 434.78 204.35 434.78 500.00
sigpreju 250.00 156.67 264.00 250.00 250.00 156.67 160.00 252.76 sigtpreju 200.00 125.33 211.20 200.00 200.00 125.33 160.00 202.21
Test 02-0158SSLLB_B91
254
Span 1 Span 2 Span 3 Span 4 Span 5 Span 6 Span 7 Span 8 g 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 q 9.00 15.00 9.00 9.00 9.00 9.00 9.00 25.00 pu 20.25 29.25 20.25 20.25 20.25 20.25 20.25 30.00
pser 14.00 20.00 14.00 14.00 14.00 14.00 14.00 G -30.00 -100.00 -50.00 -100.00 -150.00 -10.00 -10.00 -10.00 Q -30.00 -50.00 -40.00 -100.00 -100.00 -8.00 -8.00 -8.00 l 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00
Mu 91.13 131.63 91.13 91.13 91.13 91.13 91.13 135.00 Nu -85.50 -210.00 -127.50 -285.00 -352.50 -25.50 -25.50 -18.00
Mser 63.00 90.00 63.00 63.00 63.00 63.00 63.00 Nser -60.00 -150.00 -90.00 -200.00 -250.00 -18.00 -18.00 Vu 60.75 87.75 60.75 60.75 60.75 60.75 60.75 90.00
Main reinforcement calculation according to ULS Mu/A 74.03 89.63 65.63 34.13 20.63 86.03 86.03 131.40 ubu 0.161 0.100 0.049 0.059 0.036 0.125 0.094 0.083 a 0.221 0.133 0.062 0.077 0.046 0.167 0.123 0.108 z 0.410 0.426 0.439 0.436 0.442 0.420 0.428 0.430
Au 6.12 20.57 7.97 8.35 9.18 11.27 5.21 6.46 Main reinforcement calculation with prejudicial cracking according to SLS
Mser/A 51.000 60.000 45.000 23.000 13.000 59.400 59.400 0.000 a 0.4186 0.6678 0.6303 0.4737 0.4737 0.6328 0.6923 0.6157
Mrb 87.53 220.78 302.44 121.16 121.16 182.01 258.82 267.55 A 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 B -3.0000 -3.0000 -3.0000 -3.0000 -3.0000 -3.0000 -3.0000 -3.0000 C -0.4533 -0.8511 -0.3788 -0.2044 -0.1156 -0.8426 -0.8250 0.0000 D 0.4533 0.8511 0.3788 0.2044 0.1156 0.8426 0.8250 0.0000
alpha1 0.238 0.432 0.428 z 0.414 0.385 0.386
Aserp 10.22 10.99 10.75 Main reinforcement calculation with very prejudicial cracking according to SLS
Mser/A 51.00 60.00 45.00 23.00 13.00 59.40 59.40 0.00 a 0.47 0.72 0.68 0.53 0.53 0.68 0.69 0.67
Mrb 96.93 231.67 319.66 132.43 132.43 192.27 258.82 283.60 A 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 B -3.0000 -3.0000 -3.0000 -3.0000 -3.0000 -3.0000 -3.0000 -3.0000 C -0.5667 -1.0638 -0.4735 -0.2556 -0.1444 -1.0532 -0.8250 0.0000 D 0.5667 1.0638 0.4735 0.2556 0.1444 1.0532 0.8250 0.0000
alpha1 0.203 z 0.420
Asertp 14.049 Transverse reinforcement calculation
tu 0.68 0.98 0.68 0.68 0.68 0.68 0.68 1.00 k 0.57 0.40 0.00 -0.14 -0.41 0.00 0.00 0.00
At/st 1.87 7.08 4.31 3.90 4.77 7.34 3.45 4.44 Recapitulation
Aflex 6.12 20.57 7.97 10.22 14.05 11.27 10.75 6.46 e0 -0.95 -1.67 -1.43 -3.17 -3.97 -0.29 -0.29 -0.13
Aminfsimp 0.75 2.38 1.86 0.87 0.87 2.11 1.24 1.37 Aminfcomp 0.83 2.54 2.01 0.90 0.90 2.80 1.65 0.30
At 1.87 7.08 4.31 3.90 4.77 7.34 3.45 4.44 Atmin 1.60 3.40 2.00 1.60 1.60 3.40 1.60 1.60
Test 02-0158SSLLB_B91
255
Finite elements modeling
Linear elements: beams with imposed mesh
29 nodes,
28 linear elements.
2.91.4 Results sheet
Results comparison
Solver Model Units Reference RC Expert Deviation
Inf. main reinf. T1 cm2 6.12 6.12 0.00% Sup. main reinf. T1 cm2 0 0 Min. main reinf. T1 cm2 0.75 0.75 0.00% Trans. reinf. T1 cm2 1.87 1.87 0.00% Inf. main reinf. T2 cm2 20.57 20.57 0.00% Sup. main reinf. T2 cm2 0 0 Min. main reinf. T2 cm2 2.38 2.38 0.00% Trans. reinf. T2 cm2 7.08 7.08 0.00% Inf. main reinf. T3 cm2 7.97 7.97 0.00% Sup. main reinf. T3 cm2 0 0 Min. main reinf. T3 cm2 1.86 1.86 0.00% Trans. reinf. T3 cm2 4.31 4.31 0.00% Inf. main reinf. T4 cm2 10.22 10.23 0.10% Sup. main reinf. T4 cm2 0 0
CM2 Min. main reinf. T4 cm2 0.87 0.87 0.00% Trans. reinf. T4 cm2 3.90 3.90 0.00% Inf. main reinf. T5 cm2 14.05 14.05 0.00% Sup. main reinf. T5 cm2 0 0.94 Min. main reinf. T5 cm2 4.20 4.20 Trans. reinf. T5 cm2 4.77 4.77 0.00% Inf. main reinf. T6 cm2 11.27 11.27 0.00% Sup. main reinf. T6 cm2 0 0 Min. main reinf. T6 cm2 2.11 2.11 0.00% Trans. reinf. T6 cm2 7.34 7.34 0.00% Inf. main reinf. T7 cm2 10.75 10.76 0.09% Sup. main reinf. T7 cm2 0 0 Min. main. reinf. T7 cm2 1.24 1.24 0.00% Trans. reinf. T7 cm2 3.45 3.45 0.00% Inf. main reinf. T8 cm2 6.46 6.46 0.00% Sup. main reinf. T8 cm2 0 0 Min. main reinf. T8 cm2 1.37 1.37 0.00% Trans. reinf. T8 cm2 4.44 4.44 0.00%
The "Mu limit" method must be applied in order to achieve the same results.
Test 02-0162SSLLB_B91
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2.92 Test No. 02-0162SSLLB_B91: Linear element in simple bending - without compressed reinforcement
2.92.1 Description sheet
Reference: J. Perchat (CHEC) reinforced concrete course
Analysis type: static linear;
Element type: planar.
2.92.2 Overview Beam with 8 isostatic spans subjected to uniform loads.
Units
Forces: kN
Moment: kN.m
Stresses: MPa
Reinforcement density: cm2
Geometry
Beam dimensions: 0.2 x 0.5 ht
Length: l = 42 m in 7 spans of 6m,
Materials properties
Longitudinal elastic modulus: E = 20000 MPa,
Poisson's ratio: ν = 0.
Boundary conditions
Outer: o Hinged at end x = 0, o Vertical support at the same level with all other supports
Inner: Hinge z at each beam end (isostatic)
Loading
External: o Case 1 (DL): uniform linear load g = -5 kN/m (on all spans except 8) o Case 2 to 8 (LL): uniform linear load q = -9 kN/m (on spans 1, 3 to 7)
uniform linear load q = -15 kN/m (on span 2) o Case 9 (ACC): uniform linear load a = -25 kN/m (on 8th span) o Case 10 (DL): uniform linear load g = -5 kN/m (on 8th span)
Comb BAELUS: 1.35xDL+1.5xLL with duration of more than 24h (comb 101, 104 to 107) Comb BAEULI: 1.35xDL+1.5xLL with duration between 1h and 24h (comb 102) Comb BAELUC: 1.35xDL + 1.5xLL with duration of less than 1h (comb 103) Comb BAELS: 1xDL + 1*LL (comb 108 to 114) Comb BAELUA: 1xDL + 1xACC (comb 115)
Internal: None.
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Reinforced concrete calculation hypothesis:
All concrete covers are set to 5 cm
BAEL 91 calculation with the revised version 99 Span Concrete Reinforcement Application Concrete Cracking
1 B20 HA fe500 D>24h No Non prejudicial
2 B35 Adx fe235 1h<D<24h No Non prejudicial
3 B50 HA fe 400 D<1h Yes Non prejudicial
4 B25 HA fe500 D>24h Yes Prejudicial 5 B60 HA fe500 D>24h No Very
prejudicial 6 B30 Adx fe235 D>24h Yes Prejudicial 7 B40 HA fe500 D>24h Yes 160 MPa 8 B45 HA fe500 D<1h Yes Non
prejudicial
2.92.3 Reinforcement calculation
Reference solution
Span 1 Span 2 Span 3 Span 4 Span 5 Span 6 Span 7 Span 8 fc28 20 35 50 25 60 30 40 45 ft28 1.8 2.7 3.6 2.1 4.2 2.4 3 3.3 fe 500 235 400 500 500 235 500 500
teta 1 0.9 0.85 1 1 1 1 0.85 gamb 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.15 gams 1.15 1.15 1.15 1.15 1.15 1.15 1.15 1
h 1.6 1 1.6 1.6 1.6 1 1.6 1.6
fbu 11.33 22.04 33.33 14.17 34.00 17.00 22.67 39.13 fed 434.78 204.35 347.83 434.78 434.78 204.35 434.78 500.00
sigpreju 250.00 156.67 264.00 250.00 285.15 156.67 160.00 252.76 sigtpreju 200.00 125.33 211.20 200.00 228.12 125.33 160.00 202.21
g 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 q 9.00 15.00 9.00 9.00 9.00 9.00 9.00 25.00
pu 20.25 29.25 20.25 20.25 20.25 20.25 20.25 30.00 pser 14.00 20.00 14.00 14.00 14.00 14.00 14.00
l 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 Mu 91.13 131.63 91.13 91.13 91.13 91.13 91.13 135.00
Mser 63.00 90.00 63.00 63.00 63.00 63.00 63.00 Vu 60.75 87.75 60.75 60.75 60.75 60.75 60.75 90.00
Longitudinal reinforcement calculation according to ELU ubu 0.199 0.147 0.068 0.159 0.066 0.132 0.099 0.085 a 0.279 0.200 0.087 0.217 0.086 0.178 0.131 0.111 z 0.400 0.414 0.434 0.411 0.435 0.418 0.426 0.430
Au 5.24 15.56 6.03 5.10 4.82 10.67 4.91 6.28 Main reinforcement calculation with prejudicial cracking according to SLS
A 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 B -3.000 -3.000 -3.000 -3.000 -3.000 -3.000 -3.000 -3.000 C -0.56000 -0.89362 -0.87500 D 0.56000 0.89362 0.87500
alpha1 0.367 0.442 0.438 z 0.395 0.384 0.384
Aserp 6.38 10.48 10.25 Main reinforcement calculation with very prejudicial cracking according to SLS
A 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 B -3.00 -3.00 -3.00 -3.00 -3.00 -3.00 -3.00 -3.00 C -0.70 -1.60 -0.66 -0.70 -0.61371 -1.12 -0.88 0.00 D 0.70 1.60 0.66 0.70 0.61371 1.12 0.88 0.00
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Span 1 Span 2 Span 3 Span 4 Span 5 Span 6 Span 7 Span 8 alpha1 0.381
z 0.393 Asertp 7.030
Transversal reinforcement calculation tu 0.68 0.98 0.68 0.68 0.68 0.68 0.68 1.00 k 1.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00
At/st 0.69 1.79 4.31 3.45 3.45 7.34 3.45 4.44 Recapitulation
Aflex 5.24 15.56 6.03 6.38 7.03 10.67 10.25 6.28 Aminflex 0.75 2.38 1.86 0.87 1.74 2.11 1.24 1.37
At 0.69 1.79 4.31 3.45 3.45 7.34 3.45 4.44 Atmin 1.60 3.40 2.00 1.60 1.60 3.40 1.60 1.60
Finite elements modeling
Linear elements: beams with imposed mesh
29 nodes,
28 linear elements.
2.92.4 Results sheet
Results comparison
Solver Model Units Reference AD 2010 RC Expert Deviation
Inf. main reinf. T1 cm2 5.24 5.24 0.00% Sup. main reinf. T1 cm2 0 0 Min. main reinf. T1 cm2 0.75 0.75 0.00% Trans. reinf. T1 cm2 0.69 0.69 0.00% Inf. main reinf. T2 cm2 15.56 15.56 0.00% Sup. main reinf. T2 cm2 0 0 --- Min. main reinf. T2 cm2 2.38 2.38 0.00% Trans. reinf. T2 cm2 1.79 1.79 0.00% Inf. main reinf. T3 cm2 6.03 6.03 0.00% Sup. main reinf. T3 cm2 0 0 --- Min. main reinf. T3 cm2 1.86 1.86 0.00% Trans. reinf. T3 cm2 4.31 4.31 0.00% Inf. main reinf. T4 cm2 6.38 6.38 0.00% Sup. main. reinf. T4 cm2 0 0 ---
CM2 Min. main reinf. T4 cm2 0.87 0.87 0.00% Trans. reinf. T4 cm2 3.45 3.45 0.00% Inf. main reinf. T5 cm2 7.03 7.04 0.14% Sup. main reinf. T5 cm2 0 0 --- Min. main reinf. T5 cm2 1.74 1.74 0.00% Trans. reinf. T5 cm2 3.45 3.45 0.00% Inf. main reinf. T6 cm2 10.67 10.67 0.00% Sup. main reinf. T6 cm2 0 0 --- Min. main reinf. T6 cm2 2.11 2.11 0.00% Trans. reinf. T6 cm2 7.34 7.34 0.00% Inf. main reinf. T7 cm2 10.25 10.27 0.20% Sup. main reinf. T7 cm2 0 0 --- Min. main reinf. T7 cm2 1.24 1.24 0.00% Trans. reinf. T7 cm2 3.45 3.45 0.00% Inf. main reinf. T8 cm2 6.28 6.28 0.00% Sup. main. reinf. T8 cm2 0 0 --- Min. main reinf. T8 cm2 1.37 1.37 0.00% Trans. reinf. T8 cm2 4.44 4.44 0.00%
The "Mu limit" method must be applied to attain the same results.
Test 03-0206SSLLG_CM66
259
2.93 Test No. 03-0206SSLLG_CM66: Design of a Steel Structure according to CM66.
2.93.1 Data
Calculation model: Simple metallic framework with a concrete floor.
Load case: o Permanent loads: 150 kg/m² for the floor and 25kg/m² for the roof. o Overloads: 250 kg/m² on the floor. o Wind loads on region II for a normal location o Snow loads on region 2B at an altitude of 750m.
CM66 Combinations
Model preview
Test 03-0206SSLLG_CM66
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Structure’s load case Code No. Type Title CMP 1 Static SW + Dead loads CMS 2 Static Overloads for usage CMV 3 Static Wind overloads along +X in overpressure CMV 4 Static Wind overloads along +X in depression CMV 5 Static Wind overloads along -X in overpressure CMV 6 Static Wind overloads along -X in depression CMV 7 Static Wind overloads along +Z in overpressure CMV 8 Static Wind overloads along +Z in depression CMV 9 Static Wind overloads along -Z in overpressure CMV 10 Static Wind overloads along -Z in depression CMN 11 Static Normal snow overloads
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2.93.2 Effel Structure results Displacement Envelope (“CMCD" load combinations)
Envelope of linear element forces D DX DY DZ Env. Case No. Max.
location (cm) (cm) (cm) (cm) Max(D) 213 148 CENTER 12.115 0.037 12.035 -1.393 Min(D) 188 1.1 START 0.000 0.000 0.000 0.000
Max(DX) 204 72.1 START 3.138 3.099 0.434 0.244 Min(DX) 204 313 END 2.872 -1.872 -0.129 -2.174 Max(DY) 213 148 CENTER 12.115 0.037 12.035 -1.393 Min(DY) 213 61.5 END 9.986 -0.118 -9.985 0.046 Max(DZ) 201 371 CENTER 4.149 -0.006 -0.188 4.145 Min(DZ) 203 370 CENTER 4.124 -0.006 -0.240 -4.118
Envelope of forces on linear elements (“CMCFN” load combinations)
Envelope of linear element forces Fx Fy Fz Mx My Mz Env. Cas
e No. MaxSite (T) (T) (T) (T*m) (T*m) (T*m) Max (Fx) 120 4.1 START 19.423 -4.108 -1.384 -0.003 1.505 7.551 Min (Fx) 138 98 START -41.618 -0.962 -0.192 0.000 0.000 0.000 Max(Fy) 120 57 END -13.473 16.349 -0.016 -0.003 0.002 55.744 Min(Fy) 120 60 START -15.994 -16.112 -0.006 -3E-004 6E-006 53.096 Max(Fz) 177 371 START -3.486 -0.118 2.655 0.000 0.000 0.000 Min(Fz) 187 370 START -3.666 -0.147 -2.658 0.000 0.000 0.000 Max(Mx) 120 111 END 3.933 4.840 0.278 0.028 -4E-005 11.531 Min(Mx) 120 21 END -22.324 13.785 -0.191 -0.028 -0.004 42.562 Max(My) 177 371 CENTER -3.099 -0.118 -0.323 0.000 4.403 -0.500 Min(My) 179 370 CENTER -3.283 -0.155 0.321 0.000 -4.373 -0.660 Max (Mz) 120 57 END -13.473 16.349 -0.016 -0.003 0.002 55.744 Min (Mz) 120 59.2 END -19.455 -8.969 -0.702 -0.003 -0.001 -57.105 Envelope of linear element stresses (“CMCFN” load combinations)
Envelope of linear element stresses sxxMax sxyMax sxzMax sFxx sMxxMax Env. Cas
e No. MaxSite (MPa) (MPa) (MPa) (MPa) (MPa) Max(sxxMax) 120 59.2 END 273.860 -14.696 -1.024 -16.453 290.312 Min(sxxMax) 120 292 START -150.743 0.000 0.000 -150.743 0.000 Max(sxyMax) 120 57 START 262.954 37.139 -0.030 -15.609 278.562 Min(sxyMax) 120 60 END 241.643 -36.595 -0.011 -18.536 260.179 Max(sxzMax) 185 371 START -2.949 -0.183 3.876 -2.949 0.000 Min(sxzMax) 179 370 START -3.104 -0.255 -3.882 -3.104 0.000 Max(sFxx) 120 293 END 161.095 9E-005 -0.002 161.095 0.000 Min(sFxx) 120 292 START -150.743 0.000 0.000 -150.743 0.000
Max(sMxxMax) 120 59.2 END 273.860 -14.696 -1.024 -16.453 290.312 Min(sMxxMax) 1 1.1 START -4.511 3.155 -0.646 -4.511 0.000
Test 03-0206SSLLG_CM66
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2.93.3 CM66 Effel Expertise results
Hypotheses
For columns
Deflections: 1/150 Envelopes deflections calculation.
Buckling XY plane: Automatic calculation of the structure on displaceable nodes XZ plane: Automatic calculation of the structure on fixed nodes
Lateral-torsional buckling: Ldi automatic calculation: hinged restraint Lds automatic calculation: hinged restraint
For the rafters
Deflections: 1/200 Envelopes deflections calculation.
Buckling: XY plane: Automatic calculation of the structure on displaceable nodes XZ plane: Automatic calculation of the structure on fixed nodes
Lateral-torsional buckling: Ldi automatic calculation: no restraint Lds automatic calculation: hinged restraint
For the columns
Deflections: 1/150 Envelopes deflections calculation.
Buckling: XY plane: Automatic calculation of the structure on displaceable nodes XZ plane: Automatic calculation of the structure on displaceable nodes
Lateral-torsional buckling: Ldi automatic calculation: hinged restraint Lds automatic calculation: hinged restraint
Optimization parameters
Work ratio optimization between 90 and 100%
All the sections from the library are available.
Labels optimization. The results of the optimization given below correspond to an iteration of the finite elements calculation.
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Deflection verification
Ratio
Max values on the element
Columns: L / 168
Rafter: L / 96
Column: L / 924
Test 03-0206SSLLG_CM66
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CM Stress diagrams
Work ratio
Stresses
Max values on the element
Columns: 375.16 MPa
Rafter: 339.79 MPa
Column: 180.98 MPa
Test 03-0206SSLLG_CM66
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Buckling lengths
Lfy
Lfz
Test 03-0206SSLLG_CM66
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Lateral-torsional buckling lengths
Ldi
Lds
Test 03-0206SSLLG_CM66
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Optimization
2.93.4 Effel Structure / Advance Design comparison
Maximum displacement (CMCD) Effel value
(cm) Advance Design 2010 value
(cm) Deviation
(%) 12.115 12.126 0.09%
Envelope normal force (CMCFN)
Values Effel value (T)
AD 2010 value (T)
Deviation (%)
Min (Fx) -41.618 -41.629 0.03% Max (Fx) 19.423 19.466 0.2%
Envelope bending moment (CMCFN)
Element Effel value (T.m)
AD 2010 value (T.m)
Deviation (%)
Min (Mz) -57.105 -57.094 0.02% Max (Mz) 55.744 55.745 0.00%
Warning, the Mz bending moment of Effel Structure corresponds to the My bending moment of Advance Design.
Test 03-0206SSLLG_CM66
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CM deflections Element Effel value
(Ratio) AD 2010 value
(Ratio) Deviation
(%) Columns L / 168 (89%) L / 167.92 (89%) 0.05% Rafters L / 96 (208%) L / 96.13 (208%) -0.13%
Columns L / 924 (16%) L / 925.30 (16.2%) 0.14%
CM stresses Element Effel value
(MPa) AD 2010 value
(MPa) Deviation
(%) Columns 374.67 374.59 0.02% Rafters 339.74 347.46 2.20%
Columns 180.98 180.71 0.15%
Buckling lengths Element Value Effel
(m) AD 2010
(m) Deviation
(%) Columns Lfy 24.07 24.07 0.00%
Lfz 8.02 8.00 -0.25% Rafters Lfy 20.25 20.25 0.00%
Lfz 1.72 1.72 0.00% Columns Lfy 5.67 5.67 0.00%
Lfz 4.20 4.196 0.09%
Warning, the local axes in Effel Structure are opposite to those in Advance Design.
Lateral-torsional buckling lengths Element Value Effel
(m) AD 2010
(m) Deviation
(%) Columns Ldi 8.5 8.5 0.00%
Lds 8.5 8.5 0.00% Rafters Ldi 8.61 8.61 0.00%
Lds 1.72 1.72 0.00% Columns Ldi 2 2 0.00%
Lds 2 2 0.00%
Optimization Element Effel optimization AD 2010 optimization
Initial section
Rate (%)
Final section
Rate (%)
Initial section
Rate (%)
Final section
Rate (%)
Columns IPE500 159 IPE600 92 IPE500 159.4% IPE600 92.2% Rafters IPE400 145 IPE500 88 IPE400 147.9% IPE500 88.2%
Columns IPE400 77 IPE360 92 IPE400 76.9% IPE360 92.4%
Test 03-0207SSLLG_CM66
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2.94 Test No. 03-0207SSLLG_CM66: Design of a 2D portal frame
2.94.1 Data
Calculation model: 2D metallic portal frame. o Column section: IPE500 o Rafter section: IPE400 o Base plates: hinged. o Portal frame width: 20m o Columns height: 6m o Portal frame height at the ridge: 7.5m
Load case: o Permanent loads: 150 kg/m on the roof + elements self weight. o Usage overloads: 800 kg/ml on the roof
Mesh density: 1m
Model preview
Combinations
Code Numbers Type Title CMP 1 Static Permanent load + self weight CMS 2 Static Usage overloads
CMCFN 101 Comb_Lin 1.333P CMCFN 102 Comb_Lin 1.333P+1.5S CMCFN 103 Comb_Lin P+1.5S CMCD 104 Comb_Lin P+S
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2.94.2 Effel Structure Results
Ridge displacements (combination 104)
Diagram of normal force envelope
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Envelope of bending moments diagram
2.94.3 Effel Expert CM results
Main hypotheses
For columns
Deflections: 1/150 Envelopes deflections calculation.
Buckling: XY plane: Automatic calculation of the structure on fixed nodes XZ plane: Automatic calculation of the structure on fixed nodes
Ka-Kb Method Lateral-torsional buckling: Ldi automatic calculation: no restraints
Lds imposed value: 2 m
For the rafters
Deflections: 1/200 Envelopes deflections calculation.
Buckling: XY plane: Automatic calculation of the structure on fixed nodes XZ plane: Automatic calculation of the structure on fixed nodes Ka-Kb Method
Lateral-torsional buckling: Ldi automatic calculation: No restraints Lds imposed value: 1.5m
Optimization criteria
Work ratio optimization between 90 and 100%
Labels optimization (on Advance Design templates)
Test 03-0207SSLLG_CM66
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Deflection verification
Ratio
CM Stress diagrams
Work ratio
Stresses
Test 03-0207SSLLG_CM66
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Buckling lengths Lfy
Lfz
Lateral-torsional buckling lengths Ldi
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Lds
Optimization
2.94.4 Effel Structure / Advance Design comparison
Displacement at the ridge (load combination 104) Effel value
(cm) Advance Design 2010 value
(cm) Deviation
(%) 9.36 9.36 0.00%
Envelope normal forces
Element Effel value (T)
AD 2010 value (T)
Deviation (%)
Columns (min) -15.77 -15.77 0.00% Rafters (max) -1.02 -1.02 0.00%
Envelope bending moments
Element Effel value (T.m)
AD 2010 value (T.m)
Deviation (%)
Columns (min) -42.41 -42.41 0.00% Rafters (max) 42.41 42.41 0.00%
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CM deflections Element Effel value
(Ratio) AD 2010 value
(Ratio) Deviation
(%) Columns L / 438 (34%) L / 438 (34%) 0.00% Rafters L / 111 (180%) L / 111 (180%) 0.00%
CM stresses
Element Effel value (MPa)
AD 2010 value (MPa)
Deviation (%)
Columns 230.34 230.34 0.00% Rafters 458.38 458.38 0.00%
Buckling lengths
Element Value Effel (m)
AD 2010 (m)
Deviation (%)
Columns Lfy 5.84 5.84 0.00% Lfz 6 6 0.00%
Rafters Lfy 7.08 7.08 0.00% Lfz 10.11 10.11 0.00%
Warning, the local axes in Effel Structure have different orientation in Advance Design.
Lateral-torsional buckling lengths Element Value Effel
(m) AD 2010
(m) Deviation
(%) Columns Ldi 6 6 0.00%
Lds 2 2 0.00% Rafters Ldi 10.11 10.11 0.00%
Lds 1.5 1.5 0.00%
Optimization Element Effel optimization AD 2010 optimization
Initial section
Rate (%)
Final section
Rate (%)
Initial section
Rate (%)
Final Section
Rate (%)
Columns IPE500 98 - 98 IPE500 98 - 98 Rafters IPE400 195 IPE550 77 IPE400 195.1 IPE550 76.8
Test 03-0208SSLLG_BAEL91
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2.95 Test No. 03-0208SSLLG_BAEL91: Design of a concrete floor with an opening
2.95.1 Data
Calculation model: 2D concrete slab. o Slab thickness: 20 cm o Slab length: 20m o Slab width: 10m o The supports (punctual and linear) are considered as hinged. o Supports positioning (see scheme below) o 1,50m*2,50m opening => see positioning on the following scheme
Materials: o Concrete B25 o Young module: E= 36000 MPa
Load case: o Permanent loads: 100 kg/m2 o Permanent loads: 200 kg/ml around the opening o Punctual loads of 2T in permanent loads (see the following definition) o Usage overloads: 250 kg/m2
Mesh density: 0.5 m
Slab geometry
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Support positions
Positions of punctual loads
Test 03-0208SSLLG_BAEL91
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Global loading overview
Load Combinations
Code Numbers Type Title BAGMAX 1 Static Permanent loads + self weight
BAQ 2 Static Usage overloads BAELS 101 Comb_Lin Gmax+Q BAELU 102 Comb_Lin 1.35Gmax+1.5Q
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2.95.2 Effel Structure Results
SLS max displacements (load combination 101)
Mx bending moment for ULS load combination
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My bending moment for ULS load combination
Mxy bending moment for ULS load combination
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2.95.3 Effel RC Expert Results
Main hypothesis
Top and bottom concrete covers: 3 cm Slightly dangerous cracking Concrete B25 => Fc28= 25 MPa Reinforcement calculation according to Wood method. Calculation starting from non averaged forces.
Axi reinforcements
Ayi reinforcements
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Axs reinforcements
Ays reinforcements
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2.95.4 Effel Structure / Advance Design comparison
Max displacement for SLS (load combination 101) Effel value
(cm) Advance Design 2010 value
(cm) Deviation
(%) 0.176 0.175 2.85%
Mx and My bending moments for ULS (load combination 102) Magnitude Effel value
(kN.m) AD 2010 value
(kN.m) Deviation
(%) Max(Mx) 25.20 25.25 0.20% Min(Mx) -15.71 -15.68 0.19% Max(My) 31.17 31.24 0.22% Min(My) -18.79 -18.77 0.11%
Max (Mxy) 10.26 10.25 0.10% Min (Mxy) -10.14 -10.15 0.10%
Theoretic reinforcements
Magnitude Effel value (cm²)
AD 2010 value (cm²)
Deviation (%)
Axi 3.84 3.83 -0.24% Axs 3.55 3.628 2.23% Ayi 3.75 3.728 -0.57% Ays 4.53 4.619 1.97%
These values are obtained from the maximum values from the mesh.
3 Eurocodes 1 Tests description
The objective of this chapter is to validate the results from ADVANCEDesign 2010 according to Eurocodes 1.
Eurocodes 1 Tests description
286
Wind calculation
3.1 Portal frame with 11.31° angle – Example A.fto
3.1.1 Wind pressure calculation
Calculation according to French Appendix
Wind speed region: 1
Type of site: 0
Topographic factor: 1
Turbulence factor: 1
ADVANCE VALIDATION GUIDE
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ADVANCE DESIGN 2010 is set on:
Wind load cases are set on:
Eurocodes 1 Tests description
288
a) Wind speed (Vb) calculation:
Vb = Cdirection * Cseason * Vb0 = 1 * 1 * Vb0 = 22m/s
b) Site factor (Kr) calculation:
Kr = 0.19 * (z0/z0,II)^(0.07) = 0.19*(0.005/0.05)^(0.07) = 0.162
Note (see 4.3.2 (1) – French Appendix):
Terrain category Zo Zmin
0 Sea or coastal sea winds area; lakes and water course with wind over a distance of at least 5 m
0.005 1
II Open field, with or without a few isolated obstacles (trees, buildings, etc.) separated from each other by more than 40 times their height
05 2
IIIa Countryside with hedgerows, vineyards, hedged farmland, dispersed habitat
0.20 5
IIIB Urban or industrial zones, densely hedged farmland, orchards
0.5 9
IV Urban areas with at least 15% of the surface covered with buildings whose average height is greater than 15 m, forests
1 15
ADVANCE DESIGN 2010 returns: Kr = 0.16.
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c) Roughness coefficient Cr (z) calculation:
h = 5m
b = 15m
So: h < 2b
Cr (z) = Kr*ln(z/z0) = 0.162 * ln (z / 0.005) = 1.119
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French Appendix will give the same value:
ADVANCE DESIGN 2010 returns: 1.12.
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d) Average Wind Vm(z) speed calculation:
Vm(z) = Cr(z)*C0(z)*Vb = Cr(z) * 22 = 1.119 * 22 = 24.618 m/s
ADVANCE DESIGN 2010 returns: 24.58 m/s
e) Turbulence Intensity Iv(z) calculation:
kl: turbulence factor = 1
Co(z): topographic factor = 1
Iv(z) = 1/ln(z/z0) = 0.145
AD 2010 returns: 0.14
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f) Basic velocity pressure Qb calculation:
Qb = 0.5 * Ro_air * Vb² = 0.5 * 1.225 * 22² = 296.45 N/m²
Advance Design 2010 returns:
g) Peak velocity pressure Qp(z) calculation
Qp = (1+7Iv(z)) * 0.5*Ro_air * Vm² = (1+7*0.145) * 0.5*1.225 * 24.618² = 748 N/m²
Advance Design 2010 returns:744.82 N/m²
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h) Exposure factor Ce(z) calculation:
Ce (z) = Qp (z) / Qb = 748 / 296.45 = 2.52
French Appendix will give the same value:
ADVANCE DESIGN 2010 returns: 2.51
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3.1.2Cpe and Wind force calculation for front wind (Y)
Building characteristics:
h: building height = 5m
b: crosswind direction = 10m
d: parallel wind direction = 30m
e = min (b ; 2h) = 10m
h/d = 5/30 = 0.1667
Note: h, b, d and e are given by the picture below:
For each windwall: Cpe = Cpe10
Table 7.1 Recommended values of external pressure coefficients for the vertical walls of buildings on a rectangular plane:
Note: For buildings with h / d> 5, the total wind load may be based on the information from Sections 7.6 to 7.8 and 7.9.2.
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a) Windwall n° 1
Cpe calculation:
Windwall n°1: D area
h/d = 5/30 = 0.1667
So: h/d < 0.25
Table 7.1 Recommended values of external pressure coefficients for the vertical walls of buildings on a rectangular plane:
Note: For buildings with h / d> 5, the total wind load may be based on the information from Sections 7.6 to 7.8 and 7.9.2.
Cpe = Cpe10 = +0.7
ADVANCE DESIGN 2010 returns +0.7:
D
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Wind force calculation:
Wind force: F = Qp(z)*(Ce-Ci).
Note: in this example, Qp = 748 N/m².
F = 748 * 0.7 = 523.6 N/m²
Advance Design 2010 returns:
b) Windwall n° 4
Windwall n°4: E area
E
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h/d = 5/30 = 0.1667
h/d < 0.25
Table 7.1 Recommended values of external pressure coefficients for the vertical walls of buildings on a rectangular plane:
Note: For buildings with h / d> 5, the total wind load may be based on the information from Sections 7.6 to 7.8 and 7.9.2.
Cpe = Cpe10 = -0.3
ADVANCE DESIGN 2010 returns -0.3:
F = 748 * (-0.3) = -224.4 N/m²
Advance Design 2010 returns:
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c) Windwalls n° 2 and n°3
e<d:
A area
B area
C area
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h/d = 4/30 = 0.133
So: h/d < 0.25
Table 7.1 Recommended values of external pressure coefficients for the vertical walls of buildings on a rectangular plane:
Note: For buildings with h / d> 5, the total wind load may be based on the information from Sections 7.6 to 7.8 and 7.9.2.
Advance Design 2010 returns:
Wind force for area A:
F = 748 * (-1.2) = -898 N/m²
Advance Design 2010 returns:
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Wind force for area B:
F = 748 * (-0.8) = -598 N/m²
Advance Design 2010 returns:
Wind force for area C:
F = 748 * (-0.5) = -374 N/m²
Advance Design 2010 returns:
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d) Windwalls n°5 and 6
There are 4 areas: F,G,H et I:
Note: e = 10m
e/4 = 2.5m
e/10 = 1m
e/2 = 5m
Cpe depends on the angle: 11.31°.
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For 11.31°, the values are between those two lines: 5° et 15°:
Table 7.4b - External pressure coefficient for double-sided sloped roofs
Between 5° and 15°:
11.31° -1.41 -2.07 -1.3 -2 -0.64 -1.2 -0.54
F area:
Cpe10 = -1.41
Advance Design 2010 returns:-1.411
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G area:
Cpe10 = -1.3
Advance Design 2010 returns -1.3
H area:
Cpe = Cpe10 = – 0.64
Advance Design 2010 returns -0.637
I area:
Cpe = Cpe10 = -0.54
Advance Design 2010 returns -0.537:
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Wind force for area F:
F = Qp(z)*(Ce-Ci) = 748 * (-1.41) = -1055 N/m²
Fx = F * sin(alpha) = 207 N/m²
Fy = F * cos(alpha) = 1035 N/m²
Advance Design 2010 returns:
Wind force for area G:
F = 748 * (-1.30) = -972 N/m²
Fx = 191 N/m²
Fy = 953 N/m²
Advance Design 2010 returns:
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Wind force for area H:
F = 748 * (-0.637) = -476 N/m²
Fx = 93.35 N/m²
Fy = 467 N/m²
Advance Design 2010 returns:
Wind force for area I:
F = 748 * (-0.537) = -402 N/m²
Fx = 79 N/m²
Fy = 394 N/m²
Advance Design 2010 returns:
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3.1.3Cpe and Wind force calculation for parallel wind (X)
Building characteristics:
h: building height = 5m (or 4m for windwalls n° 2 and 3)
b: crosswind direction = 30m
d: parallel wind direction = 10m
e = min (b ; 2h) = 10m (or 8m for windwalls n° 2 and 3)
h/d = 5/10 = 0.5
Note: h, b, d and e are given by the picture below:
For each windwall: Cpe = Cpe10
Table 7.1 Recommended values of external pressure coefficients for the vertical walls of buildings on a rectangular plane:
Note: For buildings with h / d> 5, the total wind load may be based on the information from Sections 7.6 to 7.8 and 7.9.2.
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a) Windwall n° 3
Cpe calculation:
Windwall n°3: D area
h/d = 4/10 = 0.4
So: 0.25 < h/d < 0.5
The values will be between those two lines:
Table 7.1 Recommended values of external pressure coefficients for the vertical walls of buildings on a rectangular plane:
Note: For buildings with h / d> 5, the total wind load may be based on the information from Sections 7.6 to 7.8 and 7.9.2.
0.5 -1.2 -1.4 -0.8 -1.1 -0.5 +0.72 +1.0 -0.34
Cpe = Cpe10 = +0.720
D
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Advance Design 2010 returns:
Wind force calculation:
Wind force: F = Qp(z)*(Ce-Ci) = 748 * 0.720 = 538 N/m²
Advance Design 2010 returns: F = 511 N/m².
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b) Windwall n° 2
0.5 -1.2 -1.4 -0.8 -1.1 -0.5 +0.72 +1.0 -0.34
Note: For buildings with h / d> 5, the total wind load may be based on the information from Sections 7.6 to 7.8 and 7.9.2.
Cpe = Cpe10 = -0.34
Advance Design 2010 returns:
E
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Wind force calculation:
Wind force: F = Qp(z)*(Ce-Ci) = 748 * (-0.34) = -250 N/m²
Advance Design 2010 returns: F = 241 N/m².
c) Windwalls n° 1 and n°4
e = 5m
d = 10m
So: e<d:
There will be 2 areas: A and B:
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h/d = 5/10 = 0.5
For each area, Cpe = Cpe10.
Table 7.1 Recommended values of external pressure coefficients for the vertical walls of buildings on a rectangular plane
0.5 -1.2 -1.4 -0.8 -1.1 -0.5 +0.73 +1.0 -0.367
Note: For buildings with h / d> 5, the total wind load may be based on the information from Sections 7.6 to 7.8 and 7.9.2.
Advance Design 2010 returns:
A area
B area
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Wind force for area A:
F = 748 * (-1.2) = -898 N/m²
Advance Design 2010 returns:
Wind force for area B:
F = 748 * (-0.8) = -598 N/m²
Advance Design 2010 returns:
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3.2 Monopitch frame with 15° angle – Example B.fto 3.2.1Lateral Wind (X direction): Cpe calculation on rooftop
Building characteristics:
h: building height = 6.68m
b: crosswind direction = 30m
e = min (b ; 2h) = 13.358m
h/d = 5/10 = 0.5
Roof angle = 15°
Loads areas must be: F, G and H.
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Advance Design 2010 returns:
For each windwall: Cpe = Cpe10
Table 7.3a - External pressure coefficients applied on single-sided sloped roofs
Note: for F, G and H areas, the table will give two values for Cpe10, which means that for each area, there will be a positive and a negative to take into account.
Note: When θ = 0° (see Table a), the pressure varies rapidly between positive and negative values for an angle α ranging from + 5° to + 45°; this is the reason the positive and negative values are specified for these slopes. For these roofs, it is necessary to consider two cases: a case for all the positive values, and a case for all negative values. A mixture of positive and negative values on one side is not allowed.
F
G
F
H
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F area: Cpe neg = -0.9
Cpe pos = = +0.2
G area: Cpe neg = -0.8
Cpe pos = +0.2
H area: Cpe neg = -0.3
Cpe pos = +0.2
Advance Design 2010 returns:
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3.3 Portal frame with 10° angle – Example C.fto
3.3.1Lateral Wind (X): Cpe calculation and Wind force calculation
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The values for a 10° angle must be calculated:
Note 1: For each area, the interpolation must be done between values with same sign.
• For I area, the first value will be an interpolation between -0.6 and -0.4. The second value will be 0, because there is no other positive value to take into account.
• For J area, the first value will be an interpolation between -0.6 and -1.0. The second value will be an interpolation between +0.2 and +0.0.
Note 2: It is not allowed to have negative and positive values for the same pitch.
Advance Design 2010 returns: 3 problems
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Snow calculation
3.4 Portal frame with 11.31° angle – Example A.fto
Region of structure: A2
Altitude of site: 350m
Exposure factor: 1
Thermal factor: 1
Snow load cases are set on:
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Normal snow:s = µi • Ce • Ct • sk( 5.1 )
µi is given by:
Table 5.2 - Shape coefficients
sk,0 is given by:
Regions:
A1 A2 B1 B2 C1 C2 D E
Sk characteristic snow load value on soil at an altitude less than 200 m
0.45 0.45 0.55 0.55 0.65 0.65 0.90 1.40
SAd calculation value of exceptional snow load on soil
- 1.00 1.00 1.35 - 1.35 1.80 -
Variation law of the characteristic load for an altitude greater than 200 m
ΔS1 ΔS2
Which means that sk = 0.45 + 350/1000 – 0.20 = 0.60 kN/m²
So: Normal snow: s = µi • Ce • Ct • sk = 0.8 * 1 * 1 * 0.60 kN/m² = 0.48 kN/m²
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Advance Design 2010 returns:
Accidental snow:s = µi • sAd = 0.8 * 1 = 0.80 kN/m²( 5.1 )
Regions:
A1 A2 B1 B2 C1 C2 D E
Sk characteristic snow load value on soil at an altitude less than 200 m
0.45 0.45 0.55 0.55 0.65 0.65 0.90 1.40
SAd calculation value of exceptional snow load on soil
- 1.00 1.00 1.35 - 1.35 1.80 -
Advance Design 2010 returns:
4 Eurocodes 2 Tests description
The objective of this chapter is to validate the results from ADVANCEDesign 2010 according to Eurocodes 2.
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4.1 Test No. 01 – EC2: Minimum reinforcement of a beam
4.1.1 Overview Search of the minimum longitudinal reinforcement to be implemented on a beam. The calculation is performed on EC2, French DAN.
Units
I. S.
Geometry
Length: l = 5.50 m (5.90m between axes)
Section: b = 0.4m and h = 0.8 m.
Material properties
Concrete: C25/30
Fck= 25 N/mm²
Fyk= 500 N/mm²
Concrete density: 25 kN/m3
Boundary conditions
Hinge at end x = 0,
Simple support at extremity x = 5.50 m (horizontal restraint).
Loading
Self weight
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4.1.2 Reference results
Reference solution
Clause 7.3.2
s
cteffctcs
AFkkAσ
×××= .min,
14,0 ≤=ck (in simple bending)
65,0=k (rectangular beam with h ≥ 80 cm)
Fct.eff = Fctm = 2.56MPa
²16.02
8.04.0 mAct =×
=
50016.056.265.04.0min, ×××=sA ²13.2 cm=
Clause 9.2.1.1
dbffA tyk
ctms .26.0min, =
ctmf =2.56MPa
ykf =500MPa
tb = 0.4m
d = 0.72m
=××== 72.04.0500
56.226.0.26.0min, dbffA tyk
ctms ²84.3 cm
Therefore: ²84.3min, cmAs =
4.1.3 Results sheet
ADVANCE Design: See test 01.zip
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4.2 Test No. 02 – EC2: Longitudinal reinforcement of a beam under a linear load - horizontal level behavior law
4.2.1 Overview Research of the longitudinal reinforcement to implement on a beam. The calculation is performed on EC2, French DAN.
Units
I. S.
Geometry
Length: l = 5.50 m (5.90m between axes),
Section: b = 0.4m and h = 0.8 m.
Material properties
Concrete: C25/30
Fck= 25 N/mm²
Fyk= 500 N/mm²
Concrete density: 25 kN/m3
Boundary conditions
Hinge at end x = 0,
Simple support at extremity x = 5.50 m (horizontal restraint).
Loading
No self weight
G = 100 kN/ml
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4.2.2 Reference results
Reference solution
C25/30 concrete strength: MPaffc
ckcccd 67,16
5,125. ===
γα
ELU maximum bending moment calculation:
o mKNlPMEd .5878
²50,5)10035,1(8
².=
××==
Calculation of the reduced moment:
170,067,16²72,040,0
587,0²
=××
==cd
Ed
FbdMμ
ξ calculation: [ ] [ ] 234,0)170,021(125,1)21(125,1 =×−−=−−= μξ
zb leverage calculation: mdzb 653,0)234,04,01(72,0)4,01( =×−=−= ξ
Reinforcement bars cross section calculation: ²68,2078,434653,0
587,0.
cmzMA
sb
Edu =
×==
σ
4.2.3 Results sheet
ADVANCE Design: See test 02.zip
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326
4.3 Test No. 03 – EC2: Longitudinal reinforcement of a beam under a linear load - inclined stress strain behavior law
4.3.1 Overview Research of the longitudinal reinforcement to implement on a beam. The calculation is performed on EC2, French DAN.
Units
I. S.
Geometry
Length: l = 5.50 m (5.90m between axes),
Section: b = 0.4m and h = 0.8 m.
Material properties
Concrete: C25/30
Fck= 25 N/mm²
Fyk= 500 N/mm²
Concrete density: 25 kN/m3
Steel ductility: Class A
Boundary conditions
Hinge at end x = 0,
Simple support at extremity x = 5.50 m (horizontal restraint).
Loading
No self weight
G = 100 kN/ml
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4.3.2 Reference results Reference solution
C25/30 concrete strength: MPaffc
ckcccd 67,16
5,125. ===
γα
ELU maximum bending moment calculation:
o mKNlPMEd .5878
²50,5)10035,1(8
².=
××==
Calculation of the reduced moment: mKNlPMEd .5878
²50,5)10035,1(8
².=
××==
ξ calculation: [ ] [ ] 234,0)170,021(125,1)21(125,1 =×−−=−−= μξ
zb leverage calculation: mdzb 653,0)234,04,01(72,0)4,01( =×−=−= ξ
To determine the ultimate stress sσ of tensile reinforcement, the elongation of the tensile reinforcement must be determined.
For this, a concrete shortening of 3,5‰ is taken into account.
Therefore, we have ‰46.11234,0
234,015,315,35,3 =−
=−
=−
=ξ
ξε
xxd
S
We verify that this value remains less than ukε9,0 , or 0,9*25=22,5‰ for an A class steel.
Table 2.1 Reinforcement classes
Then we determine if it is in the plastic range:
11.46‰ ‰ 175,2200000
78,4340 <===
S
yds E
fε : is in the plastic range.
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Thus, we obtain:
05,1=k from the 2.1 table of the C annex of EC2 (class A steel)
Table 2.1 Reinforcement classes
( )( )
( ) MPakff
suk
SS
S
ykyd 63,443
)175,225()175,246.11(05,01
15,150011
0
0 =⎟⎟⎠
⎞⎜⎜⎝
⎛−
−+=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−−+=
εεεε
γ
We deduce then the section of the corresponding reinforcement:
²26,2063,443653,0
587,0.
cmzMA
sb
Edu =
×==
σ
4.3.3 Results sheet
ADVANCE Design: See test 03.zip
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4.4 Test No. 04 – EC2: Longitudinal reinforcement of a beam under a concentrated load - horizontal level behavior law
4.4.1 Overview Research of the longitudinal reinforcement to implement on a beam. The calculation is performed on EC2, French DAN.
Units
I. S.
Geometry
Length: l = 5.50 m (5.90m between axes),
Section: b = 0.4m and h = 0.8 m.
Material properties
Concrete: C25/30
Fck= 25 N/mm²
Fyk= 500 N/mm²
Concrete density: 25 kN/m3
Boundary conditions
Hinge at end x = 0,
Simple support at extremity x = 5.50 m (horizontal restraint).
Loading
No self weight
G = 150 kN and Q = 250 kN at mid-span
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4.4.2 Reference results Reference solution
C25/30 concrete strength: MPaffc
ckcccd 67,16
5,125. ===
γα
ELU maximum bending moment calculation:
o mKNlPM Ed .8524
90,5)2505,115035,1(4.
=××+×
==
Calculation of the reduced moment: 246,067,16²72,040,0
852,0²
=××
==cd
Ed
FbdMμ
ξ calculation: [ ] [ ] 359,0)246,021(125,1)21(125,1 =×−−=−−= μξ
zb leverage calculation: mdzb 617,0)359,04,01(72,0)4,01( =×−=−= ξ
Reinforcement bars cross section calculation:
²76,31²10.76,3178,434617,0
852,0.
4 cmmzMA
sb
Edu ==
×== −
σ
4.4.3 Results sheet
ADVANCE Design: See test 04.zip
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4.5 Test No. 05 – EC2: Longitudinal reinforcement of a beam under a linear load - horizontal level behavior law
4.5.1 Overview Search of reinforcement to be implemented on a beam. The calculation is performed on EC2, French DAN.
Units
I. S.
Geometry
Length: l = 6.10 m (6.40m between axes),
Section: b = 0.25m and h = 0.7 m.
Material properties
Concrete: C25/30
Fck= 25 N/mm²
Fyk= 500 N/mm²
Concrete density: 25 kN/m3
Boundary conditions
Hinge at end x = 0,
Simple support at extremity x = 6.10 m (horizontal restraint).
Loading
No self weight
G = 25 kN/ml and Q = 30 kN/ml
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4.5.2 Reference results Reference solution
C25/30 concrete strength: MPaffc
ckcccd 67,16
5,125. ===
γα
ELU maximum bending moment calculation:
o mKNlPMEd .4048
²40,6)305,12535,1(8
².=
××+×==
Calculation of the reduced moment: 244,067,16²63,025,0
404,0²
=××
==cd
Ed
FbdMμ
ξ calculation: [ ] [ ] 356,0)244,021(125,1)21(125,1 =×−−=−−= μξ
zb leverage calculation: mdzb 574,0)356,025,01(63,0)4,01( =×−=−= ξ
Reinforcement bars cross section calculation: ²19,1678,434574,0
404,0.
cmzMA
sb
Edu =
×==
σ
4.5.3 Results sheet
Thonier: See test 05.XLS
ADVANCE Design: See test 05.zip
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4.6 Test No. 06 – EC2: Transverse reinforcement of a beam subjected to a linear load
4.6.1 Overview Search of reinforcement to be implemented on a beam. The calculation is performed on EC2, French DAN.
Units
I. S.
Geometry
Length: l = 6.10 m (6.40m between axes),
Section: b = 0.25m and h = 0.7 m.
Material properties
Concrete: C25/30
Fck= 25 N/mm²
Fyk= 500 N/mm²
Concrete density: 25 kN/m3
Boundary conditions
Hinge at end x = 0,
Simple support at extremity x = 6.10 m (horizontal restraint).
Loading
No self weight
G = 25 kN/ml and Q = 30 kN/ml
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4.6.2 Reference results Reference solution
Shear force calculation: EdV
kNVEd 2522
305.12535.1=
×+×=
Shear force calculation with reduction: redEdV , : non considered by Advance Design
Concrete shear resistance: max,RdV
θθα
gtgfvzbV cdwcw
Rd cot.... 1
max, +=
o θ: connecting rod slope angle = 45° o z: internal forces lever arms. It can be z=0,9d = 0.567m
o 1=cwα in simple bending without prestress.
o 1v : cracked concrete resistance reduction coefficient of the shear force:
54.02502516,0
25016,01 =⎥⎦
⎤⎢⎣⎡ −=⎥⎦
⎤⎢⎣⎡ −= ckfv
MNgtgfvzbV cdwcw
Rd 630.011
67.1654.0567.025.01cot
.... 1max, =
+××××
=+
=θθ
α
max,RdEd VV < : no crushing of concrete struts. The section is properly sized.
Shear resistance calculation in the absence of transverse reinforcement: cRdV ,
[ ]{ } dbkvfkCMaxV wcpckLcRdcRd ..;)..100(. 1min3/1
,, σρ +=
o 12.05.118.018.0
, ===c
cRdC γ
o 0,2563.163020012001
)lim(
≤=+=+=ètresmild
k
o 02,0.
≤=db
A
w
sLLρ
²19.16 cmAsL = (see test 05)
02,001028.063.025.0
1019.16.
4
≤=××
==−
dbA
w
sLLρ
o 177.0255.1
053,0.053,0 2/12/1min =×== ck
c
fVγ
[ ]{ } MNMaxV cRd 0866.063.025.055.063.025.0177.0;)2501028.0100(563.112.0 3/1, =××=×××××=
cRdEd VV ,> : transverse reinforcement implementation
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Shear force on transverse vertical reinforcement:
mlcmfzV
sAVfz
sAV
ywd
EdswEdywd
swsRd /²22.10
178.434567.0252.0
cot..cot..., =
××=>⇒>=
θθ
4.6.3 Results sheet
ADVANCE Design: See test 06.zip
Eurocodes 2 Tests description
336
4.7 Test No. 07 – EC2: Longitudinal reinforcement of a beam under a linear load - horizontal level behavior law
4.7.1 Overview Research of the longitudinal reinforcement to implement on a beam. The calculation is performed on EC2, French DAN.
Units
I. S.
Geometry
Length: l = 5.50 m (5.80m between axes),
Section: b = 0.25m and h = 0.6 m.
Material properties
Concrete: C25/30
Fck= 25 N/mm²
Fyk= 500 N/mm²
Concrete density: 25 kN/m3
Boundary conditions
Hinge at end x = 0,
Simple support at extremity x = 5.50 m (horizontal restraint).
Loading
Self weight
G = 5 kN/ml and Q = 75 kN/ml
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4.7.2 Reference results Reference solution
C25/30 concrete strength: MPaffc
ckcccd 67,16
5,125. ===
γα
Self weight: mlkNPp /75.32560.025.0 =××=
ELU maximum bending moment calculation:
o mKNlPMEd .5238
²80,5)755,1)575.3(35,1(8
².=
××++×==
Calculation of the reduced moment: 430,067,16²54,025,0
523,0²
=××
==cd
Ed
FbdM
μ
Limited reduced moment: 372.0lim =iteμ (for Fe500)
itelimμμ > : compressed reinforcement bars implementation (or resize of the concrete section)
Tensile reinforcement calculation (section A1):
The tensile reinforcement calculation must be associated with a moment corresponding to μlim.
buFbdM
²lim
lim =μ => mMNFbdM bu .452,067,16²54,025,0372,0²limlim =×××== μ
αlim calculation: [ ] 617,0)372,021(125,1lim =×−−=α
zb leverage calculation:
Reinforcement bars cross section calculation: ²54,2578,434407,0
452,0lim1 cm
FzM
Aedb
=×
==
Compressed reinforcement calculation (section A’)
Compressed reinforcement stress calculation:
o ( ) ( ) 00000
000
018.3'03.054.0617.0
54.0617.05.3'5.3
=−××
=−= ddd lu
lusce α
αε
o MPafEf
cdsces
cdsce 78.43418.2
20000078.434
000 ==⇒==> σε
Compressed reinforcement calculation: ²20,378.434)03,054,0(
452,0523,0)'(
' cmddMM
Asc
ulu =−
−=
−−
=σ
A2 reinforcement bars calculation section to balance A’: ²20,378,43478.43403,2'2 cm
fAA
cd
sce ===σ
Eurocodes 2 Tests description
338
Total section to implement:
The total section to implement is:
A=A1+A2=28,74cm² in the inferior part (tensile reinforcement)
A’=3,20cm² in the superior part (compressed reinforcement)
4.7.3 Results sheet
ADVANCE Design: See test 07.zip
5 Eurocodes 3 Tests description
The objective of this chapter is to validate the results from ADVANCEDesign 2010 according to Eurocodes 3.
Eurocodes 3 Tests description
340
5.1 Test No. 01 – EC3: 2D frame Design
5.1.1 Data
Calculation model: 2D metallic portal frame o Column section: HEA200 o Beam section: IPE240 o Base plates: fixed. o Portal frame width: 10m o Columns height: 3m o Portal frame height at the ridge: 4m
Load case: o Gravitational load: 15kN
Model preview
Material properties
Steel S355: fy = 355N/mm2, ε = (235/fy)0.5 = 0.81
ADVANCE VALIDATION GUIDE
341
Cross sections
1. Beam
EC3 Iy Iz Iw It A Sy Sz Wel,y Wel,z U.M. cm4 cm4 cm6 cm4 cm2 cm2 cm2 cm3 cm3
3892.00 283.6 37.39 12.88 39.12 24.83 19.14 324.00 47.30
2. Column
EC3 Iy Iz Iw It A Sy Sz Wel,y Wel,z U.M. cm4 cm4 cm6 cm4 cm2 cm2 cm2 cm3 cm3
3692.0 1336.0 108.0 20.98 53.88 41.59 18.08 388.6 133.6
Eurocodes 3 Tests description
342
5.1.2 Axial forces – N
5.1.3 Shear forces – T
5.1.4 Bending moments – M
ADVANCE VALIDATION GUIDE
343
5.1.5 Classification of the beam cross section (IPE 240)
Flange:
( )⇒=⋅>=
− 29.7981.58.9
2/2.6120 ε class 1
Web:
5.047.000.190.012<=−>−=−
⋅⋅
= αψ forfAN
y
Ed
⇒⋅
=<=α
ε3604.6270.30wtc
class 1
1. Reference – the class of cross section = max(class of flange; class of web) = CLASS 1.
2. AD 2010 – CLASS 1.
Eurocodes 3 Tests description
344
5.1.6 Resistance of the beam cross section (IPE 240)
Compression
The design value of the compression force NEd, should satisfy:
kNNwhereNN
EdRdc
Ed 55.67,00.1,
=≤
The design resistance of cross section for uniform compression Nc,Rd, should be:
00.105.076.1388
55.67
76.138800.1
103551012.39
sec32,1
,
34
0,
0,
≤==⇒
=⋅⋅⋅
=⋅
=
⋅=⇒−
−
Rdc
Ed
M
yRdc
M
yRdc
NN
kNfA
N
fANtioncrossorclassfor
γ
γ
AD 2010:
Shear
The design value of shear force VEd, should satisfy:
kNVwhereVV
EdRdc
Ed 49.64,00.1,
=≤
where Vc,Rc is the design shear resistance:
( )
( )
00.116.029.392
49.64
29.39200.11
3103551014.193/
14.19
40.1614.1922
,
34
0,
2
22
≤==⇒
=⋅⋅⋅⋅
=⋅
=
=
>⇒⋅⋅>⋅⋅++⋅⋅−=
−
Rdpl
Ed
M
yvRdpl
v
wwfwfv
VV
kNfA
V
cmA
cmcmthtrttbAA
γ
η
AD 2010:
ADVANCE VALIDATION GUIDE
345
Bending moment
The design value of the bending moment MEd, should satisfy:
kNmMwhereMM
EdRdc
Ed 91.92,00.1,
=≥
Mc,Rd – the design resistance for bending moment about one principal axis is:
00.171.014.13091.92
14.1301000.1
355060.366
60.366,sec21
,
4,
3
0,
<==⇒
=⋅⋅
=
=−⋅
=
−
Rdc
Ed
Rdc
plM
yplRdc
MM
kNmM
cmWwheretioncrossorclassforfW
Mγ
AD 2010:
Lateral torsional buckling
( )( ) ( ) ( )
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⋅−⋅+
⋅⋅⋅⋅⋅
+⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅⋅⋅
⋅= ggz
t
z
w
w
zcr zCzC
IEIGLk
II
kk
LkIECM 2
222
22
21 ππ
( )
( )
( )
01324.060.283
7559.3
00.0
107559.34
046051.0/1008.8
88.12
9877.225
10.500.1
60.283/101.2
642
22
2
27
4
2
24
28
==
=
⋅=−⋅
=
=⋅⋅
⋅⋅⋅⇒
⎪⎭
⎪⎬⎫
⋅=
=
=⋅
⋅⋅⇒
⎪⎪⎭
⎪⎪⎬
⎫
===
=
⋅=
z
w
g
fzw
z
tt
z
w
z
II
mz
cmthI
I
mIEIGLk
mkNG
cmI
kNLkIE
mLkk
cmImkNE
π
π
Eurocodes 3 Tests description
346
C1 and C2 coefficients:
[ ] kNmNmM cr 07.1545.154075046051.001324.017.22598780.2 ==+⋅⋅⋅=
Slenderness for lateral torsional buckling, LTλ :
( ) ( )acurveforx
MfW
LTLT
cr
yyplLT
72.0919.0
919.01007.154
35506.3664
,
==⇒
=⋅
⋅=
⋅=
λ
λ
Design buckling resistance moment:
00.1981.070.9392.91
70.931000.1
35506.36672.0
,
4
1,,
≤==⇒
=⋅⎟⎠⎞
⎜⎝⎛ ⋅⋅
=⋅⋅= −
Rdb
Ed
M
yyplLTRdb
MM
kNmf
WxMγ
AD 2010:
MM =1
MM ⋅=ψ22010
,51.080.2
2009:
53.065.2
524.091.928
10.5158
521.091.9244.48
44.4891.92
2
1
2
1
22
2
1
ADwithobtainedvaluesthewithcontinue
willncalculatiotheCC
ADObs
CC
MLq
kNmMMkNmMM
⎩⎨⎧
==
⇒
⎩⎨⎧
==
⇒
−=⋅⋅
=⋅⋅
=
−=−=⇒⎭⎬⎫
=⋅=−==
μ
ψψ
ADVANCE VALIDATION GUIDE
347
Buckling resistance:
( )
( )
( )
( )⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≤⋅
⋅+⋅
≤⋅
⋅+⋅
==Δ=Δ⇒==
≤Δ+
⋅+⋅
Δ+⋅+
⋅
≤Δ+
⋅+⋅
Δ+⋅+
⋅
62.600.1
61.600.1
:
00.000.000.0
62.600.1
61.600.1
1
,
,
1
1
,
,
1
,,,
1
,
,,
1
,
,,
1
1
,
,,
1
,
,,
1
M
RkyLT
Edyzy
M
Rkz
Ed
M
RkyLT
Edyyy
M
Rky
Ed
EdzRdzRdyNzNy
M
Rkz
RdzEdzzz
M
RkyLT
RdyEdyzy
M
Rkz
Ed
M
Rkz
RdzEdzyz
M
RkyLT
RdyEdyyy
M
Rky
Ed
Mx
MkNx
N
Mx
MkNx
N
becomeonsverificatithe
MsiMMeefor
MMM
kMx
MMkNx
N
MMM
kMx
MMkNx
N
γγ
γγ
γγγ
γγγ
Characteristic resistance of the critical cross section
kNmWfM
kNAfN
yplyRky
yRk
14.1301060.366355
76.13881012.393553
,,
2
=⋅⋅=⋅=
=⋅⋅=⋅=
Buckling about axis y-y:
( ) ( )( )
( )
kNN
xN
bcurvexNfA
kNN
cmImkNE
L
IEN
mL
M
RkyRdby
yycr
yy
ycr
y
ycr
yycr
ycr
01.111100.1
76.138880.0
80.0672.005.3074
103551012.39
05.307410.5
1000.3892101.214.3
00.3892/101.2
10.5
1,
34
,
2
882
,
4
28
2,
2
,
,
=⋅=⋅=
=⇒=⋅⋅⋅
=⋅
=
=⋅⋅⋅⋅
=⇒
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
=
⋅=
⋅⋅=
=
−
−
γ
λ
π
Buckling about axis z-z:
( ) ( )( )
( )
kNNxN
ccurvexNfA
kNN
cmImkNE
LIEN
mL
M
RkzRdbz
yycr
yy
zcr
z
zcr
zzcr
zcr
314.20800.1
76.138815.0
15.0488.2314.224
103551012.39
314.22410.5
10284101.214.3
00.284/101.2
10.5
1,
34
,
2
882
,
4
28
2,
2
,
,
=⋅=⋅=
=⇒=⋅⋅⋅
=⋅
=
=⋅⋅⋅⋅
=⇒
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
=
⋅=
⋅⋅=
=
−
−
γ
λ
π
Eurocodes 3 Tests description
348
The kyy and kzy coefficients – are calculated with “Alternative method 1”:
`60.01
1
1
1
,
,
z
y
zy
ycr
Ed
zmLTmyzy
yy
ycr
Ed
ymLTmyyy
ww
CNN
CCk
CNN
CCk
⋅⋅⋅−
⋅⋅=
⋅−
⋅⋅=
μ
μ
50.150.1562.130.4790.7350.1131.1
00.3246.366
,
,
,
, =⇒≥===≤=== zzel
zplz
yel
yply w
WW
wandWW
w
732.030114.015.01
30114.01
314.22455.6715.01
314.22455.671
1
1
995.002197.080.01
02197.01
05.307455.6780.01
05.307455.671
1
1
,
,
,
,
=⋅−
−=
⋅−
−=
⋅−
−=
=⋅−
−=
⋅−
−=
⋅−
−=
zcr
Edz
zcr
Ed
z
ycr
Edy
ycr
Ed
y
NNx
NN
NNx
NN
μ
μ
Ncr,T and Ncr,TF :
( ) ( ) ( ) ( )
kNN
NNIII
NNNNII
IN
kNN
mI
mI
mA
mzAIII
LIEIG
IA
N
TFcr
Tcrzcrzy
TcrzcrTcrzcrzy
TFcr
Tcr
w
t
g
szy
Tcr
wt
gTcr
25.199
42
64.1108
107559.3
1088.12
101.39
104.9806
,
,,0
2,,,,
0,
,
68
48
24
4820
,
2
0,
=⇒
⎥⎦
⎤⎢⎣
⎡⋅⋅
+⋅−+−+⋅
+⋅=
=⇒
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
⎬
⎫
⋅=
⋅=
⋅=
⋅=⋅++=
⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅⋅+⋅⋅=
−
−
−
−
π
ADVANCE VALIDATION GUIDE
349
Calculation of 0λ and zyyymLTmy CCCC ,,, coefficients:
( )
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
≥
⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛−
⋅=
=
⋅+
⋅⋅−+=
⇒
>=⇒
⎟⎠⎞
⎜⎝⎛ −⋅⎟
⎠⎞
⎜⎝⎛ −⋅⋅=⎟
⎟⎠
⎞⎜⎜⎝
⎛−⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛−⋅⋅
00.1
11
11
20.0276.0
25.19955.671
314.22455.67180.220.01120.0
,,
2
0
00
0
44
,,1
TFcr
Ed
zcr
Ed
LTmymLT
mzmz
LTy
LTymymymy
TFcr
Ed
zcr
Ed
NN
NN
aCC
CC
a
aCCC
NN
NNC
ε
ε
λ
Cmy,0 :
MM =1
MM ⋅=ψ2
( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
==
==
====
≥=−==⋅=
⇒
=⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−
⋅
⋅⋅⋅+=⇒
⋅−=−=
=
−=−=⇒⎭⎬⎫
=⋅=−==
−
714.0;9345.0
;224.1;003.1
00.0;00.0;0486.0/
0997.01;05.4
013.111
1081.481.4
91.92
521.091.9244.48
44.4891.92
1
,
,
,,2
2
0,
2
2
1
zyyy
mLTmy
LTLTMRk
Edpl
y
tLT
yelEd
Edyy
ycr
Ed
Edy
xymy
x
x
CC
CC
dbNNn
IIa
WA
NM
NN
xMLIE
C
mcm
kNmMkNmMM
kNmMM
γ
ε
δπ
δ
ψψ
Eurocodes 3 Tests description
350
Calculation of kyy and kzy:
67.050.113.160.0
714.01
05.307455.671
732.0224.1003.160.01
1
336.19345.01
05.307455.671
995.0224.1003.11
1
,
,
=⋅⋅⋅−
⋅⋅=⋅⋅⋅−
⋅⋅=
=⋅−
⋅⋅=⋅−
⋅⋅=
z
y
zy
ycr
Ed
zmLTmyzy
yy
ycr
Ed
ymLTmyyy
ww
CNNCCk
CNNCCk
μ
μ
Buckling resistance:
( )
( ) 00.198.0
00.114.13072.0
91.9267.0
00.176.138815.0
55.67:61.6
00.137.1
00.114.13072.0
91.92336.1
00.176.138880.0
55.67:61.6
1
,
,
1
1
,
,
1
<=⋅
⋅+⋅
=⋅
⋅+⋅
>=⋅
⋅+⋅
=⋅
⋅+⋅
M
RkyLT
Edyzy
M
Rkz
Ed
M
RkyLT
Edyyy
M
Rky
Ed
Mx
MkNx
N
Mx
MkNx
N
γγ
γγ
AD 2010:
ADVANCE VALIDATION GUIDE
351
5.1.7 Classification of the column cross section (HEA 200)
Flange:
( )⇒=⋅>=
− 29.79510
2/5.6200 ε class 1
Web:
5.045.000.192.012
<=−>−=−⋅
⋅= αψ for
fAN
y
Ed
⇒⋅
=<=α
ε368.6415.26wtc
class 1
1. Reference – the class of cross section = max(class of flange; class of web) = CLASS 1.
2. AD 2010 – CLASS 3.
Eurocodes 3 Tests description
352
5.1.8 Resistance of the column cross section (HEA 200)
Compression
The design value of compression force NEd, should satisfy:
kNNwhereNN
EdRdc
Ed 49.76,00.1,
=≤
The design resistance of cross section for uniform compression Nc,Rd, should be:
00.104.0965.191049.76
965.191000.1
103551083.53
sec32,1
,
34
0,
0,
≤==⇒
=⋅⋅⋅
=⋅
=
⋅=⇒−
−
Rdc
Ed
M
yRdc
M
yRdc
NN
kNfA
N
fANtioncrossorclassfor
γ
γ
AD 2010:
Shear
The design value of shear force VEd, should satisfy:
kNVwhereVV
EdRdc
Ed 59.53,00.1,
=≤
where Vc,Rc is the design shear resistance:
( )( )
00.1145.057.37059.53
57.37000.11
3103551008.183/
08.1822
,
34
0,
2
≤==⇒
=⋅⋅⋅⋅
=⋅
=
=⋅⋅++⋅⋅−=−
Rdpl
Ed
M
yvRdpl
fwfv
VV
kNfA
V
cmtrttbAA
γ
AD 2010:
ADVANCE VALIDATION GUIDE
353
Bending moment
The design value of the bending moment MEd, should satisfy:
kNmMwhereMM
EdRdc
Ed 91.92,00.1,
=≥
Mc,Rd – the design resistance for bending moment about one principal axis is determined as follows:
00.161.047.15291.92
47.1521000.1
355050.429
50.429,1
,
4,
3
0,
<==⇒
=⋅⋅
=
=−⋅
=
−
Rdc
Ed
Rdc
plM
yplRdc
MM
kNmM
cmWwhereclassforfW
Mγ
AD 2010:
Lateral torsional buckling
( )( ) ( ) ( )
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⋅−⋅+
⋅⋅⋅⋅⋅
+⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
⋅⋅⋅
⋅= ggz
t
z
w
w
zcr zCzC
IEIGLk
II
kk
LkIECM 2
222
22
21 ππ
( )
( )
( )
2424
642
22
2
27
4
2
24
28
1000.8100.8100.1336108216.10
00.0
108216.104
005510.0/1008.8
98.20
685.3076
00.300.100.1336
/101.2
mcmII
mz
cmthI
I
mIEIGLk
mkNG
cmI
kNLkIE
mLkk
cmImkNE
z
w
g
fzw
z
tt
z
w
z
−⋅==⋅
=
=
⋅=−⋅
=
=⋅⋅
⋅⋅⋅⇒
⎪⎭
⎪⎬⎫
⋅=
=
=⋅
⋅⋅⇒
⎪⎪⎭
⎪⎪⎬
⎫
===
=
⋅=
π
π
Eurocodes 3 Tests description
354
C1 and C2 coefficients
[ ] kNmM cr 455.992005510.00081.01685.307657.2 =+⋅⋅⋅=
Slenderness for lateral torsional buckling, LTλ :
( ) ( )acurveforx
MfW
LTLT
cr
yyplLT
951.0406.0
406.010455.922
355050.4294
,
==⇒
=⋅
⋅=
⋅=
λ
λ
Design buckling resistance moment:
00.164.084.14491.92
84.1441000.1
35505.42995.0
,
4
1,,
≥==⇒
=⋅⎟⎠⎞
⎜⎝⎛ ⋅⋅
=⋅⋅= −
Rdb
Ed
M
yyplLTRdb
MM
kNmf
WxMγ
AD 2010:
MM =1
MM ⋅= ψ2
⎩⎨⎧
==
⇒
⎩⎨⎧
==
⇒
−=−=⇒⎭⎬⎫
=⋅=−==
00.057.2
2010:
00.057.2
730.091.9287.67
87.6791.92
2
1
2
1
2
1
CC
ADObs
CC
kNmMMkNmMM
ψψ
ADVANCE VALIDATION GUIDE
355
Buckling resistance:
( )
( )
( )
( )⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≤⋅
⋅+⋅
≤⋅
⋅+⋅
==Δ=Δ⇒==
≤Δ+
⋅+⋅
Δ+⋅+
⋅
≤Δ+
⋅+⋅
Δ+⋅+
⋅
62.600.1
61.600.1
:
00.000.000.0
62.600.1
61.600.1
1
,
,
1
1
,
,
1
,,,
1
,
,,
1
,
,,
1
1
,
,,
1
,
,,
1
M
RkyLT
Edyzy
M
Rkz
Ed
M
RkyLT
Edyyy
M
Rky
Ed
EdzRdzRdyNzNy
M
Rkz
RdzEdzzz
M
RkyLT
RdyEdyzy
M
Rkz
Ed
M
Rkz
RdzEdzyz
M
RkyLT
RdyEdyyy
M
Rky
Ed
Mx
MkNx
N
Mx
MkNx
N
becomeonsverificatithe
MsiMMeefor
MMM
kMx
MMkNx
N
MMM
kMx
MMkNx
N
γγ
γγ
γγγ
γγγ
Characteristic resistance of the critical cross section:
kNmWfM
kNAfN
yplyRky
yRk
4725.152105.42910355
74.19121088.531035563
,,
43
=⋅⋅⋅=⋅=
=⋅⋅⋅=⋅=−
−
Buckling about axis y-y:
( ) ( )( )
( )
kNNxN
ccurvexNfA
kNN
cmImkNE
LIE
N
mL
M
RkyRdby
yycr
yy
ycr
y
ycr
yycr
ycr
96.164400.1
74.191286.0
86.0474.033.8502
103551088.53
33.850200.3
1000.3692101.214.3
00.3692/101.2
00.3
1,
34
,
2
882
,
4
28
2,
2
,
,
=⋅=⋅=
=⇒=⋅⋅⋅
=⋅
=
=⋅⋅⋅⋅
=⇒
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
=
⋅=
⋅⋅=
=
−
−
γ
λ
π
Eurocodes 3 Tests description
356
Buckling about axis z-z:
( ) ( )( )
( )
kNNxN
bcurvexNfA
kNN
cmImkNE
LIEN
mL
M
RkzRdbz
yycr
yy
zcr
z
zcr
zzcr
zcr
3.139600.1
74.191273.0
73.0789.0566.3073
103551088.53
566.307300.3
101336101.214.3
00.1336/101.2
00.3
1,
34
,
2
882
,
4
28
2,
2
,
,
=⋅=⋅=
=⇒=⋅⋅⋅
=⋅
=
=⋅⋅⋅⋅
=⇒
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
=
⋅=
⋅⋅=
=
−
−
γ
λ
π
The kyy and kzy coefficients– are calculated with “Alternative method 1”:
z
y
zy
ycr
Ed
zmLTmyzy
yy
ycr
Ed
ymLTmyyy
ww
CNNCCk
CNNCCk
⋅⋅⋅−
⋅⋅=
⋅−
⋅⋅=
60.01
1
1
1
,
,
μ
μ
50.150.1525.160.13380.20350.110.1
60.3885.429
,
,
,
, =⇒≥===≤=== zzel
zplz
yel
yply w
WW
wsiWW
w
975.0024886.073.01
024886.01
566.307349.7673.01
566.307349.761
1
1
998.0008996.086.01
008996.01
33.850249.7686.01
33.850249.761
1
1
,
,
,
,
=⋅−
−=
⋅−
−=
⋅−
−=
=⋅−
−=
⋅−
−=
⋅−
−=
zcr
Edz
zcr
Ed
z
ycr
Edy
ycr
Ed
y
NNx
NN
NNx
NN
μ
μ
Ncr,T:
kNN
mI
mI
mA
mzAIII
LIEIG
IA
N
Tcr
w
t
g
szy
Tcr
wt
gTcr
68.2279
108216.10
1098.20
1088.53
1067.9890
,
68
48
24
4820
,2
2
0,
=⇒
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
⎬
⎫
⋅=
⋅=
⋅=
⋅=⋅++=
⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅⋅+⋅⋅=
−
−
−
−
π
ADVANCE VALIDATION GUIDE
357
Calculation of 0λ and zyyymLTmy CCCC ,,, coefficients
( )
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
≥
⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛−
⋅=
=
⋅+
⋅⋅−+=
⇒
>=⇒
⎟⎠⎞
⎜⎝⎛ −⋅⎟
⎠⎞
⎜⎝⎛ −⋅⋅=⎟
⎟⎠
⎞⎜⎜⎝
⎛−⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛−⋅⋅
00.1
11
11
20.0316.0
68.227949.761
566.307349.76157.220.01120.0
,,
2
0
00
0
44
,,1
TFcr
Ed
zcr
Ed
LTmymLT
mzmz
LTy
LTymymymy
Tcr
Ed
zcr
Ed
NN
NN
aCC
CC
a
aCCC
NN
NN
C
ε
ε
λ
Cmy,0 :
MM =1
MM ⋅= ψ2
( )
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
==
==
≥=−===
===⋅=
⇒
=⋅−⋅+⋅+=⇒
−=−=⇒⎭⎬⎫
=⋅=−==
987.0;00.1
;001.1;989.0
0994.01;04.0/
0.0;0.0;104.4
945.033.036.021.079.0
73.091.9285.67
85.6791.92
1
,
,
,0,
2
1
zyyy
mLTmy
y
tLT
MRk
Edpl
LTLTyelEd
Edyy
ycr
Ediimy
CC
CC
II
aNN
n
dbWA
NM
NN
C
kNmMMkNmMM
γ
ε
ψψ
ψψ
Eurocodes 3 Tests description
358
Calculation of kyy and kzy:
507.05.11.16.0
987.01
33.850249.761
975.0001.1989.06.01
1
997.000.11
33.850249.761
998.0001.1989.01
1
,
,
=⋅⋅⋅−
⋅⋅=⋅⋅⋅−
⋅⋅=
=⋅−
⋅⋅=⋅−
⋅⋅=
z
y
zy
ycr
Ed
zmLTmyzy
yy
ycr
Ed
ymLTmyyy
ww
CNNCCk
CNNCCk
μ
μ
Buckling resistance:
( )
( ) 0.138.0
00.14725.15295.0
91.92507.0
00.174.191273.0
49.76:61.6
0.168.0
00.14725.15295.0
91.92997.0
00.174.191286.0
49.76:61.6
1
,
,
1
1
,
,
1
<=⋅
⋅+⋅
=⋅
⋅+⋅
<=⋅
⋅+⋅
=⋅
⋅+⋅
M
RkyLT
Edyzy
M
Rkz
Ed
M
RkyLT
Edyyy
M
Rky
Ed
Mx
MkNx
N
Mx
MkNx
N
γγ
γγ
AD 2010:
ADVANCE VALIDATION GUIDE
359
5.2 Example 1 - Class of cross section (EC3)
5.2.1 Overview Determinate the class of the bellow cross section subjected to uniform compression. The calculation is performed on EC3.
Material properties
Steel S355: fy = 355N/mm2, ε = (235/fy)0.5 = 0.81
Cross section properties
A = 174cm2;
Iy = 1.179 · 105 cm4;
Iz = 2.145 · 104 cm4;
Iw = 8.93 · 106 cm6
Wy,el = 2885cm3;
Wy,el = 5321cm3
Cross section stresses
Eurocodes 3 Tests description
360
Top flange
The compressed top flange is composed by two plates perpendicular on the web, with uniform compression stresses.
Class 4
Bottom flange
Class 4
34.111467.1318
2/)8500(=⋅>=
−= εtc
my case
34.111417.1212
2/)8300(=⋅>=
−= εtc
ADVANCE VALIDATION GUIDE
361
Web
The Web is a single plate, with uniform compression stresses.
Class 4
The class of cross section = max (class top flange; class bottom flange; class web) = CLASS 4.
In AD : The class of cross section -- CLASS 4.
02.3442758
600=⋅>== ε
tc
my case
Eurocodes 3 Tests description
362
5.3 Example 2 - Class of cross section (EC3)
5.3.1 Overview Determinate the class of the bellow cross section subjected to strong axis bending. The calculation is performed on EC3.
Material properties
Steel S355: fy = 355N/mm2, ε = (235/fy)0.5 = 0.81
Cross section properties
A = 174cm2;
Iy = 1.179 · 105 cm4;
Iz = 2.145 · 104 cm4;
Iw = 8.93 · 106 cm6
Wy,el = 2885cm3;
Wy,el = 5321cm3
Cross section stresses
ADVANCE VALIDATION GUIDE
363
Top flange
The compressed top flange is composed by two plates perpendicular on the web, with uniform compression stresses.
Class 4
my case
34.111467.1318
2/)8500(=⋅>=
−= εtc
Eurocodes 3 Tests description
364
Web
The class of cross section = max (class top flange; class web) = CLASS 4. In AD : The class of cross section -- CLASS 4.
Class 1
my case
7.8536758
6005.034.0
188.188.1 12
=⋅
<==<=
−<−=⇒⋅−=
αεα
ψσσ
tc
ADVANCE VALIDATION GUIDE
365
5.4 Example 3 - Class of cross section (EC3)
5.4.1 Overview Determinate the class of the bellow cross section subjected to strong axis bending. The calculation is performed on EC3.
If the cross section is a symmetric I:
Cross section stresses
Top flange
Class 4
34.111467.13
182/)8500(
=⋅>=−
= εtc
Eurocodes 3 Tests description
366
Web
Web it’s a single plate, with bending stresses.
OBS: The class of cross section = max (class top flange; class web) = CLASS 4.
In AD : The class of cross section -- CLASS 4.
44.100124758
60023.67835.0
100.1 12
=⋅<==<=⋅=
−=⇒⋅−=
εεα
ψσσ
tc
my case
Class 3
ADVANCE VALIDATION GUIDE
367
5.5 Example 4: Tension column – design value of the resistance (EC3)
Determine the design value of the resistance to tension force, knowing the following:
Material properties
Steel S235: fy = 235N/mm2; fu = 360N/mm2;
Cross section properties (IPE 200)
A = 28.48cm2;
Iy = 1943 cm4;
Iz = 142.40 cm4
Design value of resistance to normal force:
⎪⎪⎩
⎪⎪⎨
⎧
=
⎪⎪⎭
⎪⎪⎬
⎫
==⋅⋅
=⋅⋅
==⋅
=⋅
= kNkNdaNfA
kNdaNfA
N
M
unet
M
y
Rdt 28.66920.73873820
25.1360048.289.09.0
28.6696692800.1
235048.28
min
2
0,
γ
γ
In AD 2010 => Nt,Rd = 669.28kN
Eurocodes 3 Tests description
368
5.6 Example 5: Tension column – design value of the resistance (EC3)
Determine the design value of the resistance to tension force, knowing the following:
Material properties
Steel S235: fy = 235N/mm2; fu = 360N/mm2;
Cross section properties (D20/1.5)
A = 87.18cm2;
Iy = 3754.15 cm4;
Iz = 3754.15 cm4;
Design value of resistance to normal force:
⎪⎪⎩
⎪⎪⎨
⎧
=
⎪⎪⎭
⎪⎪⎬
⎫
==⋅⋅
=⋅⋅
==⋅
=⋅
= kNkNdaNfA
kNdaNfA
N
M
unet
M
y
Rdt 73.20487.2259225970
25.1360018.879.09.0
73.204820487300.1
235018.87
min
2
0,
γ
γ
In AD 2010 => Nt,Rd = 2048.71kN
031009-0409-0914
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